Infinity and intensionality (completed)

About the project

The project “Infinity and intensionality: Towards a new Synthesis” investigates two clusters of questions so far only studied in isolation.

1. The theory of infinite sets, which is the standard foundation for nearly all of today’s mathematics, but which is threatened by paradoxes.
2. Intensional notion such as propositions, properties, and relations. Propositions are a central concern in linguistics, psychology, and philosophy.

Our overarching hypothesis is that real progress requires a new synthesis, where the two clusters of questions are tackled in a unified way. To address the first cluster, we supplement set theory with intensional notions of collection, number, and generality.  

To develop satisfactory theories of intensional notions, we draw on concepts and theories developed by critics of infinitary set theory. 

Having run from August 2021, the project finished at the end of 2025.

Read our final report (pdf)

Goals

The project’s goals are: 

• to develop a novel and distinctive philosophy of mathematics, which retains the theory of infinite sets but supplements it with intensional notions of collection, number, and generality
• to use concepts and methods from the critical views of infinity to develop new approaches to intensional phenomena, e.g., propositions and properties
• to offer a novel account of “large” totalities used in semantics and mathematics by developing an alternative notion of set
• to reshape the hotly debated hierarchical conception of reality by bringing to bear intensional conceptions of collection and generality
• to transfer ideas from the critical views of infinity to Cantor’s absolute infinity, thus retaining the strength of Cantorian set theory while incorporating insights from the critical views. 

Our final report (pdf) describes how these goals were achieved. 

Duration

01.08.21 - 31.12.25.

Financing

This project is funded by the Norwegian Research Council (NFR) with the project number 314435.

Outreach

Video: Agustín Rayo

Watch Agustín Rayo (MIT)  summarize in 3 minutes his talk at the Infinity and Intensionality Colloquium, June 2022.   

Video: Neil Barton

In this short video, Neil Barton explains the motivation for the conference Engineering the Concept of Collection, 20-21 June 2022

Podcast:  Øystein Linnebo

Listen to the podcast Universitetspodden: "Finnes universets begynnelse?" (in Norwegian) on the origins of the universe with guests Øystein Linnebo and astrophysicist Trygve Leithe Svalheim.

Events

Infinity and Intensionality Colloquium

Peter Fritz - Nonconservative Extensions by Propositional Quantifiers and Modal Incompleteness

Time and place: June 12, 2024 3:00 PM–4:30 PM, Georg Morgenstjernes hus, rom 452

Propositional modal logics can be extended by propositional quantifiers, i.e., quantifiers binding proposition letters understood as variables. This talk investigates whether such an extension is always conservative. It is shown that the answer depends on the way in which propositional quantifiers are added. On a minimal approach, according to which propositional quantifiers only have to satisfy the classical principles of quantification, every classical modal logic has a conservative extension by propositional quantifiers. However, in the context of normal modal logics it is natural to require propositionally quantified extensions also to be closed under the rule of necessitation. It is shown that in this setting, there are normal modal logics whose propositionally quantified extensions are nonconservative. Nonconservativity is shown to be a special case of a number of model-theoretic notions of incompleteness. More tentatively, it is suggested that nonconservativity indicates incompleteness in a more substantial sense, concerning the intended target of capturing the logics of modalities.

4:30-6pm: The Oslo group continues the discussion with Peter Fritz.

Organiser

Øystein Linnebo

Infinity and Intensionality Colloquium: Gabriel Uzquiano (University of Southern California)

Gabriel Uzquiano (University of Southern California) will give a talk titled "Classes, and their Potential Parts"

Time and place: Oct. 3, 2023 3:00 PM–4:30 PM, Georg Morgenstiernes hus, room 652 and on Zoom

Classes, and their Potential Parts

Abstract

We explore the prospects of a mereological account of proper classes within a potentialist framework. Actualism conceives of the set-theoretic hierarchy as a completed or actual multiplicity of sets and urelements, which form the domain of a binary relation governed by the Zermelo-Fraenkel axioms. The moral of Russell’s paradox, for the actualist, is that not all multiplicities form a set, e.g., there is no set of all and only non-self-membered sets, but there is no problem in principle to paraphrase apparent singular reference to proper classes in terms of plural reference to sets. On the other hand, Potentialism regards the existence of a set as potential relative to that of its members. So, even if there is no set of all and only actual non-self-membered sets, there can be one, which is not yet actual. There is no vantage point from which to conceive of the set-theoretic hierarchy as a completed or actual multiplicity of sets and urelements; no matter what sets may be actual, there can be further sets that are not yet actual. In the absence of a completed multiplicity of sets, there is no room for a plural interpretation of proper classes. We look at a mereological interpretation of proper classes as an alternative. We attempt to make sense of classes, whether proper or improper, as sums of singletons in a potentialist framework. We will, in particular, motivate a view of proper classes as nonrigid sums, which, unlike sets, change their parts as the set-theoretic hierarchy progressively unfolds. While conceived as urelements, proper classes lack elements, they nevertheless have singletons as atomic proper parts, which allows one to simulate membership. What makes proper classes distinctive on the present framework is the fact that there is no vantage point from which its parts have been completely given. Whatever their actual parts may be, they can have further parts that are not yet actual.

The talk will be hybrid (in-person and on Zoom).

Infinity and Intensionality Colloquium: Kit Fine (New York University)

Kit Fine (University Professor and Silver Professor of Philosophy and Mathematics, New York University) will give a talk titled "The Case for Numerical and Mereological Indeterminacy".

Time and place: Sep. 13, 2023 4:00 PM–5:30 PM, Georg Morgenstiernes hus, room 652 and on Zoom

The Case for Numerical and Mereological Indeterminacy

Abstract

I argue for the possibility of numerical and mereological indeterminacy under a global, rather than a local, conception of vagueness.

The talk will be hybrid (in-person and on Zoom).

Infinity and Intensionality Colloquium: Joel David Hamkins (University of Notre Dame)

Joel David Hamkins (O’Hara Professor of Philosophy and Mathematics, University of Notre Dame) will give a talk titled "Varieties of potentialism".

Time and place: Apr. 25, 2023 3:30 PM–5:00 PM, On Zoom

Abstract

I shall survey the surprisingly enormous variety of potentialist conceptions, even in the case of arithmetic potentialism, spanning a spectrum from linear inevitabilism and other convergent potentialist conceptions to more radical nonamalgamable branching-possibility potentialist conceptions. Underlying the universe-fragment framework for potentialism, one finds a natural modal vocabulary capable of expressing fine distinctions between the various potentialist ideas, as well as sweeping potentialist principles. Similarly diverse conceptions of ultrafinitism grow out of the analysis. Ultimately, the various convergent potentialist conceptions, I shall argue, are implicitly actualist, reducing to and interpreting actualism via the potentialist translation, whereas the radical-branching nonamalgamable potentialist conception admits no such reduction.

The talk will be held on Zoom.

Infinity and Intensionality Colloquium: Sam Roberts (University of Konstanz)

Sam Roberts (University of Konstanz) will give a talk titled "Actualism and Potentialism"

Time and place: Jan. 25, 2023 3:30 PM–5:00 PM, On Zoom

Abstract

In this talk I aim to get clear on the debate between actualism and potentialism in the philosophy of set theory. I’ll start by clarifying the positions. It's natural to see actualism as something like the negation of potentialism. I’ll argue, however, that this is wrong. Instead, we should view it as the claim that our best theory of sets is a theory of the actual sets. I’ll then go on to outline what I take to be the main considerations for and against each view. Although I won’t come down in favour of either, I hope to help clarify what’s at issue between them.

The talk will be held on Zoom.

Infinity and Intensionality Colloquium: Joan Rand Moschovakis (Occidental College, emerita)

Joan Rand Moschovakis (Occidental College, emerita) will give a talk titled "Brouwer's Notion of Choice Sequence and its Descendants"

Time and place: Dec. 7, 2022 4:00 PM–5:30 PM, On Zoom

Abstract 

In his 1907 dissertation Brouwer sketched the construction of the lawlike real numbers, showing that they had the same order type as the rationals. He then claimed that a (denumerable) ``scale'' of this kind could be completed to a measurable continuum by using a primitive notion of continuity or ``fluidity.''

By 1918 however, he was willing to accept infinitely proceeding sequences of integers or rationals, and structured sets of these sequences, as admissible mathematical objects. Arbitrary Cauchy sequences of rationals represented (uncountably many) real numbers. Continuity, no longer primitive, was eventually retained as a necessary attribute of every total function on the continuum.

Lawlike sequences are surely intensional objects. Arbitrary choice sequences should be purely extensional. Brouwer's notions of ``spread'' and ``fan,'' the basic structures of intuitionistic analysis, enabled successive restrictions on choices; but Brouwer avoided higher-order restrictions.

Beginning around 1950 Kleene, Vesley, Kreisel, Troelstra, and Veldman clarified, interpreted, and adapted Brouwer's notions. The axiomatic systems they proposed to characterize arbitrary, lawlike, ``lawless'' and other varieties of choice sequence became objects of study in their own right. In this talk I will focus on Troelstra's many contributions to the theory of choice sequences.

The talk will be held on Zoom.

Agustín Rayo (MIT) -- "Why I'm not an Absolutist"

Agustín Rayo (MIT) will give a talk titled "Why I'm not an Absolutist"

Time and place: June 15, 2022 3:00 PM–4:30 PM, Georg Morgenstiernes hus, Room 652 and on Zoom

Abstract

I consider certain limitations of Absolutism -- the view that there is sense to be made of absolutely general quantification -- and sketch an alternative proposal.

The talk will be hybrid (in-person and on Zoom).

Stewart Shapiro (Ohio State University and University of Oslo) - "Does God collapse potential infinity to actual infinity?"

Stewart Shapiro (Ohio State University and University of Oslo) will give a talk titled "Does God collapse potential infinity to actual infinity?"

(joint work with Samuel Levey and Øystein Linnebo)

Time and place: Apr. 27, 2022 3:45 PM–5:15 PM, On Zoom

Abstract

Beginning with Aristotle, and continuing well into the nineteenth century, major philosophers and mathematicians rejected the notion of the actual infinite. They argued that the only sensible notion is that of potential infinity. Closely related to the notion of potential infinity here is that of potential existence. For Aristotle, the points interior to a line segment only exist potentially—they are places where the line can be broken.

It is, we suggest, natural to explicate this notion of potentiality in terms of modal logic. Potentiality, understood modally, is acting here as a kind of guard-rail against paradox—initially the well-known Zeno paradoxes.

The theme of this paper is to see how well this modal conception of potential infinity stands up when a Judeo-Christian deity enters the picture. If God “sees” (or otherwise comprehends) all the points there are in a given line—all of the places where the line could be cut—then these points are on the line even before the cuttings take place. If God “sees” something, it must be there; it must be real or actual in some robust sense.

The nineteenth and early twentieth centuries saw a definitive change in orientation towards the infinite, culminating in the pioneering work of Georg Cantor, who showed us how to make mathematical sense of completed infinite collections or sets. Cantor argued for the exact opposite of the traditional view, stating that the potentially infinite is dubious unless it is somehow backed by an actual infinity.

At least on the surface, however, Cantor was not entirely consistent in his rejection of the potential infinite. Sometimes he ascribed to the so-called “absolutely infinite”, or what he dubbed “inconsistent multitudes” (e.g., the ordinals), features closely analogous to those of the potentially infinite. And, often, the discussion is theological.

The situation for the Cantorian is starkly different from that of the traditional view that rejects all actual infinities. For the Cantorian, the invocation of the deity does not merely threaten to collapse the potentially infinite into the actually infinite; it threatens inconsistency. Even for God, a set of all sets, all cardinal numbers, or all ordinals, is a contradiction.

Let an extending operation be an operation that maps a plurality of objects to a larger (in some sense) such plurality. For example, take some (ordinal) numbers and produce the least upper bound of them. This is Cantor’s “second principle of number generation.”

Extending operations pose a threat if one desires to have collections that cannot be increased. Cantor talks about the multiplicity of all sets, the extended number sequence, and the like (albeit with the qualification that these totalities are “inconsistent”). If one of these totalities could serve as an argument of an extending operation, that would yield yet another object of the relevant kind, and we would have a contradiction: we would have shown that an alleged totality is not a totality after all.

At least initially, potentiality provides an appealing way to deny this availability and thus defuse the threat. For the problematic cases, at least, the extending operation cannot be applied because there is no time or possible world at which the entire plurality of objects that would serve as an argument is available. Instead, the objects that would collectively make up the desired argument are “spread out” with respect to some dimension of increase (time, modality). This is the sense in which potentiality is a “guard-rail” against paradox.

But it is dangerous to bring God into the picture. Surely, from God’s point of view, all of the objects that would collectively make up the disputed argument of an extending operation—say, all points on a line segment, or all sets, or all (ordinal) numbers—must be available. What then prevents God from applying the extending operation, even if we mortals cannot?

We shall find efforts to install a second guard-rail by denying that the extending operation can be applied—some absolute limit on its scope. Not even God can form a single collection of all the relevant objects or produce a least upper bound. Galileo, Leibniz, and their contemporary heirs try versions of this strategy. A better option is to articulate the relevant modality in a way that it becomes plausible that even for God, there is no index that has all of the relevant arguments. This sets up a framework that allows the collapse to be resisted at its very introduction.

The talk will be held on Zoom.

Michael Rathjen (University of Leeds) - "Completeness: Turing, Schütte, Feferman"

Michael Rathjen (University of Leeds) will give a talk titled "Completeness: Turing, Schütte, Feferman"

Time and place: Feb. 16, 2022 3:00 PM–4:30 PM, On Zoom

Abstract

Progressions of theories along paths through Kleene's O, adding the consistency of the previous theory at every successor step, can deduce every true Pi01-statement. This was shown by Turing in his 1938 thesis who called these progressions ordinal logics.

In 1962 Feferman proved the amazing theorem that progressions based on the uniform reflection principle can deduce every true arithmetic statement. In contrast to Turing's, Feferman's proof is very complicated, involving several cunning applications of self-reference via the recursion theorem.

Using Schütte's method of search trees (or decomposition trees) for omega-logic and reflexive induction, however, one can give a rather transparent proof.

The main aim of the talk is to present a general proof-theoretic machinery, based on search trees, and discuss intensional and extensional aspects of metamathematical results.

The talk will be held on Zoom.

Jon Erling Litland (University of Texas at Austin) - "Generating Propositions"

Jon Erling Litland (University of Texas at Austin) will give a talk titled "Generating Propositions"

Time and place: Jan. 26, 2022 3:00 PM–4:30 PM, On Zoom

Abstract

I attempt to develop a theory of structured propositions where propositions and propositional constituents like properties, relations, logical operations, and quantifiers are generated in a never-ending series of stages. The guiding idea behind the theory is that propositional constituents are individuated in terms of the contribution they make to how propositions formed by applying them are (metaphysically) grounded. I will pay particular attention to how properties that are generated at some stage can apply to entities that are generated at later stages.

The talk will be held on Zoom.

Sara Negri (University of Genoa) - "Bridges between classical and constructive reasoning for infinitary logic"

Sara Negri (University of Genoa) will give a talk titled "Bridges between classical and constructive reasoning for infinitary logic"

Time and place: Dec. 1, 2021 3:00 PM–4:30 PM, On Zoom

Bridges between classical and constructive reasoning for infinitary logic

A survey will be presented on the bridges between classical and constructive reasoning in infinitary logic. At the level of probability we show how the method of conversion of axioms into rules gives a natural approach to Glivenko sequent classes for geometric theories [1, 2]. At the structural level, we show that enriching the syntax of sequents with labels for neighbourhood semantics gives an intuitionistic calculus that shares all the properties of its classical counterpart [3] and that allows to extend to the infinitary setting the modal interpretation of intuitionistic logic [4].

References

[1] Fellin, G, Negri, S. and Orlandelli, E. Glivenko classes and constructive cut elimination in infinitary logic, ms., 2021

[2] Negri, S. Geometric rules in infinitary logic. In O. Arieli and A. Zamansky (eds) Arnon Avron on Semantics and Proof Theory of Non-Classical Logics, Outstanding Contributions to Logic, Springer, pp. 265–293, 2021.

[3] Tesi, M., Negri, S. Neighbourhood semantics and labelled calculus for intuitionistic infinitary logic. Journal of Logic and Computation, 2021, exab040, https://doi.org/10.1093/logcom/exab040.

[4] Tesi, M., Negri, S. Infinitary modal logic and the G ̈odel-McKinsey-Tarski embedding, submitted.

The talk will be held on Zoom.

Dag Normann (UiO) - "Generalising the concept of being finite"

Dag Normann (UiO) will give a talk titled "Generalising the concept of being finite"

Time and place: Oct. 21, 2021 10:15 AM–12:00 PM, On Zoom

Abstract

In the proceedings of Logic Colloquium ’69, Georg Kreisel published a seminal paper on reasons for generalising recursion theory (now computability theory), and one purpose stressed by Kreisel was that such generalisations give us the opportunity to understand which properties of being finite are used in mathematical arguments.

In this non-technical talk we will survey parts of generalised computability theory with a focus on how various properties of finite sets were generalised in infinitary settings.

This talk is joint with the Oslo Logic Seminar. The talk will take place on Zoom.

Infinity and Intensionality Seminar

Guest lecture by Andrew Arana (Université de Lorraine and CNRS)

“An introduction to purity of methods”

Time and place: Oct. 9, 2024 3:00 PM–4:30 PM, GM667

“An introduction to purity of methods”

A proof of a theorem can be said to be pure if it draws only on what is ``close'' or ``intrinsic'' to that theorem. In this talk, we will introduce the apparent preference for pure proofs that has persisted in mathematics since antiquity, alongside a competing preference for impurity. After giving an example from number theory and a brief history of purity in mathematics, we will distinguish four types of purity, based on different measures of distance between theorem and proof. We will then discuss reasons for preferring pure proofs, for the varieties of purity constraints that we have presented.

Stewart Shapiro guest lecture

Time and place: Sep. 24, 2024 2:15 PM–3:45 PM, GM652

Stewart Shapiro (OSU), "Semantic indeterminacy, concept sharpening, and set theories"

Infinity and Intensionality Seminar: Markus Pantsar (RWTH Aachen University)

Markus Pantsar (RWTH Aachen University) will give a talk titled "Recognising artificial mathematical intelligence"

Time and place: Aug. 30, 2023 4:00 PM–5:15 PM, Georg Morgenstiernes hus, room 652 and on Zoom

Recognising artificial mathematical intelligence

Abstract

The study of intelligence in animals has two potential pitfalls. First, the setting for experiments may not be suitable for animals to exhibit their characteristic intelligence. Second, the observed behaviour may be misrepresented in terms of intelligence. Often this involves ascribing unjustifiably high level of intelligence to animals. In this talk, my focus is on artificial intelligence (AI) and how its study could avoid related pitfalls. In particular, I focus on artificial mathematical intelligence. Related to the first pitfall, I ask whether mathematical AI applications are systematically deprived of the chance to exhibit their characteristic intelligence (if any). Related to the second pitfall, I ask whether there is a danger in misrepresenting the processing of current mathematical AI applications in terms of intelligence. I argue that unlike in animal studies, the second pitfall is significantly more prominent than the first one. However, I argue that the first pitfall may become more serious as mathematical AI applications reach a sufficiently high level. As a proposed solution to both present and future problems, I propose a community-based approach to recognising artificial mathematical intelligence.

The talk will be hybrid in Oslo and on Zoom.

Infinity and Intensionality Seminar: Bokai Yao (University of Notre Dame)

Bokai Yao (University of Notre Dame) will give a talk titled "Reflection with Absolute Generality"

Time and place: June 20, 2023 3:30 PM–5:00 PM, Georg Morgenstiernes hus, Room 452

Reflection with Absolute Generality

Abstract

Traditionally, reflection principles in set theory claim that the set-theoretic universe is indescribable. It is natural to consider reflection principles with absolute generality, which asserts that the universe containing everything, including sets and urelements, is indescribable. In the first part of this talk, I will consider the first-order reflection principle in urelement set theory. With the Axiom of Choice, first-order reflection holds just in case the sets of urelements are arranged in a certain way, and this equivalence falls apart without AC. In the second part of this talk, I will present my joint work with Joel Hamkins on second-order reflection principles with urelements. A standard version of second-order reflection, due to Paul Bernays, is often considered as a weak large cardinal axiom in set theory. With abundant urelements, however, Bernays’ second-order reflection principle bi-interprets a supercompact cardinal.

The talk will be hybrid in Oslo and on Zoom.

Infinity and Intensionality Seminar: Sara Negri (University of Genoa)

Sara Negri (University of Genoa) will give a talk titled:

"The development of Gentzen sequent calculus: Ketonen's discovery of invertible rule systems"

Time and place: Feb. 7, 2023 3:30 PM–5:00 PM, Georg Morgenstiernes hus, room 652 and on Zoom

The development of Gentzen sequent calculus: Ketonen's discovery of invertible rule systems

Oiva Ketonen (1913 — 2000) was the closest to a student the creator of modern proof theory Gerhard Gentzen had. Their encounter took place in 1938 — 39 in Göttingen, with Ketonen hoping to receive a suitable topic for a doctoral dissertation and Gentzen instead deeply immersed in attempts at proving the consistency of analysis. Ketonen’s thesis of 1944, his only work in logic, introduced what is today called the G3-sequent calculus. It is his best-known discovery, a sequent calculus for classical propositional logic the logical rules of which are all invertible. Few read his thesis, the results of which were instead made available through a long review by Paul Bernays. Ketonen’s calculus is the basis of Evert Beth’s tableau method and of the sequent calculi in Stephen Kleene’s influential Introduction to Metamathematics. A second result was a sharpening of the midsequent theorem, by which the number of quantifier inferences with eigenvariables could be minimized. The existence of a weakest possible midsequent followed, in the sense that if any midsequent is derivable, a weakest one is. Turning this into a contrapositive, Ketonen found a purely syntactic method for proofs of underivability that he applied to affine plane geometry. His result, in modern terms, was a positive solution to the word problem for the universal fragment of plane affine geometry, with a syntactic proof of underivability of the parallel postulate from the rest of the affine axioms as a corollary.

The talk will be hybrid in Oslo and on Zoom.

Infinity and Intensionality Seminar: Ansten Klev (Czech Academy of Sciences)

Ansten Klev (Czech Academy of Sciences) will give a talk titled "Aczel's V and the notion of set"

Time and place: Jan. 11, 2023 3:30 PM–5:00 PM, Georg Morgenstiernes hus, Room 652 and On Zoom

Aczel's V and the notion of set

When philosophers discuss the relation between set theory and type theory, they will usually have in mind simple type theory and, in particular, a hierarchy of function types built over one or two ground types. The type structure of Martin-Löf type theory differs significantly from that of simple type theory. There is, for instance, an indefinite range of ground types, each of which, moreover, is inductively generated (under a suitably general notion of inductive generation). Much light was shed on the relation between set theory and Martin-Löf type theory by Peter Aczel's construction of a model — often called V — of the constructive set theory CZF inside the latter. The construction also has more general significance for the philosophy of set theory — or so it will be argued in this talk. I will first attempt to explain the model construction in a way that is accessible also to those who are unfamiliar with Martin-Löf type theory. Thereafter I will draw some philosophical consequences pertaining to the notion of set and its relation to the notion of type. Intensionality will turn out to play a significant role in this comparison.

The talk will be hybrid in Oslo and on Zoom.

Workshops

Workshop on Modality and indefinite expensibility

Is it possible to quantify over absolutely everything? According to the extendability argument, it is not. However many objects we quantify over, it is possible to use these objects to define another object that must, on pain of paradox, lie beyond the objects over which we quantified. The topic of this mini-workshop is whether the modal operators too are subject to extendability arguments. The idea would be that however large a space of possibilities is used to interpret our modal operators, it is possible to use this space to define yet further possibilities that must lie beyond that space.

Several philosophers have recently argued that modality too is subject to a form of indefinite extendability (Justin Clarke-Doane, Kit Fine, Michael Glanzberg, Agustin Rayo, Alex Roberts, James Studd). Other philosophers reject all forms of extendability arguments (T. Williamson). An intermediate view holds that, although every extensional domain (a set or a plurality) can be extended, there are absolutely inextendable intensional domains (Florio, Linnebo).

• 12:30 - 13:45: Peter Fritz
• 14:00 - 15:15: Øystein Linnebo
• 15:30 - 16:45: Agustin Rayo

Workshop on Critical Views of Logic

Time and place: Aug. 16, 2024 1:00 PM–5:30 PM, Georg Morgenstjernes hus, rom 652

The workshop marks the publication of an anthology of recent work on "critical views of logic", edited by Mirja Hartimo, Frode Kjosavik, and Øystein Linnebo.

This book examines positions that challenge the Fregean logic-first view. It raises critical questions about logic by examining various ways in which logic may be entangled with mathematics and metaphysics.

Is logic topic-neutral and general? Can we take the application of logic for granted? This book suggests that we should not be dogmatic about logic but ask similar critical questions about logic as those Kant raised about metaphysics and mathematics. It challenges the Fregean logic-first view according to which logic is fundamental and hence independent of any extra-logical considerations. Whereas Quine assimilated logic and mathematics to the theoretical parts of empirical science, the present volume explores views that stop short of his thoroughgoing holism but instead take logic to be answerable to or entangled with some particular disciplines. The contributions provide views that assign primacy to mathematical reasons, Kantian metaphysical grounds, Husserlian transcendental phenomenological reflection, or normative considerations about how terms ought to be defined in various fields of empirical science or mathematics. Space is thereby carved out between a Fregean position on the one hand and Quinean holism on the other.

Program

• 1:00 - 1:10: Welcome and introduction
• 1:10 - 2:00: Mirja Hartimo, "Husserl's and Frege's attacks on psychologism - two views on logic, reason, and logical analysis" (30" presentation followed by 20" discussion)
• 2:10 - 3:00: Frode Kjosavik, “Kant, constructionism, and the basis for a critical view of logic” (30" presentation followed by 20" discussion)
• 3:10 - 4:00: Øystein Linnebo, “Critical plural logic–what is it and why use it?” (30" presentation followed by 20" discussion)
• 4:10 - 4:30: Peter Fritz: Critical reflections on critical views
• 4:40 - 5:00: Ole Hjortland: Critical reflections on critical views
• 5:10 - 5:30: Discussion

This will be a hybrid in-person/Zoom event.

Organiser

Øystein Linnebo

Workshop on Modality and indefinite extensibility

Time and place: Aug. 14, 2024 12:30 PM–5:00 PM, Georg Morgenstjernes hus, rom 652

Is it possible to quantify over absolutely everything? According to the extendability argument, it is not. However many objects we quantify over, it is possible to use these objects to define another object that must, on pain of paradox, lie beyond the objects over which we quantified. The topic of this mini-workshop is whether the modal operators too are subject to extendability arguments. The idea would be that however large a space of possibilities is used to interpret our modal operators, it is possible to use this space to define yet further possibilities that must lie beyond that space.

Several philosophers have recently argued that modality too is subject to a form of indefinite extendability (Justin Clarke-Doane, Kit Fine, Michael Glanzberg, Agustin Rayo, Alex Roberts, James Studd). Other philosophers reject all forms of extendability arguments (T. Williamson). An intermediate view holds that, although every extensional domain (a set or a plurality) can be extended, there are absolutely inextendable intensional domains (Florio, Linnebo).

Program

12:30 - 13:45: Peter Fritz, Gaifman-Hales and Indefinite Extensibility

I consider a view of propositions on which they form a free Boolean algebra. This can be motivated using ideas from logical atomism, along the lines of Wittgenstein, according to which every proposition is a truth-functional combination of elementary propositions. To accommodate quantifiers, it is natural to consider only complete Boolean algebras. However, in this setting there are no infinite free algebras, due to a result of Gaifman and Hales. One way of understanding this result is as establishing a kind of indefinite extensibility of propositions, which naturally leads further to an indefinite extensibility of modalities.

14:00 - 15:15: Øystein Linnebo, “Absolute modality: a progress report”

Abstract: I first explore some parallels between modality and the debate about absolutely general quantification, paying special attention to the question of indefinite extensibility. Then I attempt to extend my conception of “absolute but indefinite generality” from the case of quantification to that of modality.

15:30 - 16:45: Agustin Rayo, A Pragmatic Account of the Liar Paradox

Abstract: Accounts of the Liar Paradox are often aimed at "taming" the natural language truth predicate, by isolating a consistent core and going on to regiment and systematize that core using a formal theory. I take a different approach. I hope to show that an untamed truth predicate can be used successfully in conversation, provided that one is careful about the sorts of contexts in which one deploys it. Using an untamed truth predicate in the wrong contexts leads to trouble, but the trouble in question is does not call for a revisionary semantics for "true". Instead, Liar phenomena are best thought of instances of a broader phenomenon of communicative failures in which the conversation's presuppositions are misaligned.

Organiser

Øystein Linnebo

Generative Metaphysics and the Philosophy of Mathematics

Time and place: Mar. 2, 2023–Mar. 3, 2023, Lady Margaret Hall, University of Oxford

Description

By ‘Generative Metaphysics’, we mean approaches that construe abstract objects of various kinds as possible entities that are in some sense ‘generated’ by applying operations to other (possibly concrete) things. Such ideas have been given a fair amount of attention in the literature recently, especially with applications in philosophy of mathematics. This workshop will bring together active researchers in this field to present and discuss their work. 

Questions to be addressed include:

• How do generative approaches contrast with more traditional `declarative' approaches?
• What are the relationships between different kinds of generative approach and modality in metaphysics and mathematics?
• Are there special challenges for generative approaches to overcome?
• Are there links between generative approaches and set-theoretic potentialism?
• What are the relationships between generative approaches and notions like computation, predicativism, and definability?

Speakers

Neil Barton (Universitetet i Oslo), Sharon Berry (Indiana University Bloomington), Kit Fine (New York University), Øystein Linnebo (Universitetet i Oslo), Jon Litland (University of Texas at Austin), Eileen Nutting (University of Kansas), Sam Roberts (Universität Konstanz), Ethan Russo (New York University), Chris Scambler (University of Oxford), James Studd (University of Oxford).

Registration

The workshop will be in-person only. Attendance is free but registration is required to attend. Registration is now closed, please email the organisers if you would still like to attend.

Program (all times are UTC)

2nd March, Mary O'Brien Room, Lady Margaret Hall

• 10:00-10:30 Coffee
• 10:30-11:45 James Studd `Properties for potentialists'
• 11:45-13:00 Lunch
• 13:00-14:15 Sharon Berry `Ur-Elements and the multiverse'
• 14:15-14:30 Break
• 14:30-15:45 Sam Roberts `A potential hierarchy of properties'
• 15:45-16:00 Break
• 16:00-17:15 Øystein Linnebo `Abstraction and grounding'
• 19:30 Dinner, All Souls College

3rd March, Mary O'Brien Room, Lady Margaret Hall

• 10:00-10:30 Coffee
• 10:30-11:45 Kit Fine `Procedural Postulation: Its Application to Categoricity and Structuralism'
• 11:45-13:15 Lunch
• 13:15-14:30 Neil Barton, Ethan Russo, and Chris Scambler `Make It So: Imperatival Foundations for Mathematics'
• 14:30-14:45 Break
• 14:45-16:00 Eileen Nutting `Potentialism and hierarchies of language'
• 16:00-16:15 Break
• 16:15-17:30 Jon Litland `Essentialist Potentialism'

Abstracts

James Studd (University of Oxford) `Properties for Potentialists'

Russell’s paradox refutes naive property theory in much the same way it refutes naive set theory. In the case of sets, potentialists defend a modal response. The paradox is avoided, without ad hoc limits on set formation, by maintaining that (in a suitable, ‘interpretational’ sense of ‘can’) any plurality of items can form a set. And standard, ZF, set theory is recovered via a modal axiomatization of the iterative conception. This talk outlines a parallel response to Russell’s paradox in the case of properties. A potentialist theory of properties is framed in a modal language with both metaphysical and interpretational modal operators. An application of this theory is a contingentist-friendly way to encode ‘sets’ of incompossible items and a potentialist-friendly way to encode the theory’s intended Kripke model.

Sharon Berry (Indiana University Bloomington) `Ur-Elements and the multiverse'

Hamkins proposes a multiverse approach to pure set theory, on which there is not a unique intended hierarchy of sets but rather a number of different set-theoretic universes. I will suggest that generalizing this proposal to an account of set theory with ur-elements (as used by philosophers offering logical regimentations of scientific theories and explanations) raises some special problems --- or at least questions.

Sam Roberts (Universität Konstanz) `A potential hierarchy of properties'

According to set theoretic potentialism, set existence is a matter of co-existence. If you want to know whether there could be a set with some given elements, you just need to figure out whether those elements can all exist together. The claim, in other words, is that there could be a set of the Fs precisely when the Fs can all co-exist. With the right background assumptions, this claim gives rise to a rich universe of possible sets: enough to witness the axioms of ZFC. In this talk, I will present an analogous account of untyped properties according to which property existence is a matter of definability. If you want to know whether there could be a property with some given application conditions, you just need to figure out whether those application conditions can be defined. The claim, in other words, is that there could be the property of being F precisely when being F can be defined. I will show that with the right background assumptions, this claim gives rise to a rich universe of possible properties: enough to witness the full typed comprehension schema.

Øystein Linnebo (Universitetet i Oslo) `Abstraction and grounding' (incorporating joint work with Louis deRosset)

Are there “thin objects” whose existence does not make any substantial demand on the world? In my Thin Objects (OUP 2018) I defend an affirmative answer based on a Fregean form of abstraction. This talk develops two central aspects of the resulting view in a new and alternative way, using the notion of grounding. First, every truth about some thin objects is grounded in a truth that does not involve these objects. Second, the notorious “bad company problem” for abstractionism can be solved by insisting that every truth about the desired abstracta be grounded in a truth that is solely about some prior and independently available ontology. The resulting view is a form of “soft platonism” with some broadly Aristotelian features.

Kit Fine (New York University) `Procedural Postulation: Its Application to Categoricity and Structuralism'

I shall provide an outline of procedural postulationism and show how a certain aspect of the way it generates mathematical objects can be used to provide a unified proof of categoricity for a wide range of different theories and can also be used to circumvent the problem of accounting for indiscernibles (such as i and -i) under a structuralist conception of mathematics.

Neil Barton (Universitetet i Oslo), Ethan Russo (New York University), and Chris Scambler (University of Oxford) `Make It So: Imperatival Foundations for Mathematics'

This article articulates and assesses an imperatival approach to the foundations of mathematics. After Fine, we call the program `procedural postulationism'. The core idea for the program is that mathematical domains of interest can fruitfully be viewed as the outputs of construction procedures. Fine argues that this viewpoint has various potential epistemic and ontological benefits in philosophy of mathematics; and in any case we think the guiding idea is interesting enough to warrant close inspection. We offer a logic for `creative' imperatives---imperatives that command the introduction of new objects into the domain---and show how to treat the concept of indefinite iteration in that logic. We then give consistency proofs for arithmetic and set theory starting from the formalized hypothesis that certain commands statable in the resulting logic are executable. Using this framework, we will assess whether the view can claim to have epistemic and ontological benefits over standard `declarative' approaches.

Eileen Nutting (University of Kansas) `Potentialism and hierarchies of language'

Øystein Linnebo’s dynamic abstractionism is a potentialist account on which, e.g., the iterative hierarchy of sets is given by successive stages of application of an abstraction principle like Plural Law V. Ordinary existential and universal claims are given modal readings; the existential quantifier, for example, becomes ‘at some possible stage, there is an x’. Furthermore, the abstraction principles involved in dynamic abstraction involve two-sorted languages; the language available on the identity half of the abstraction principle is an expansion of the language on the equivalence relation half. The result is that a progression of languages must accompany the progression of stages of dynamic abstraction. Existential claims in set theory require more than just an object at a possible stage; they require the possible existence of a language that can be used in the abstraction principle that yields the objects at the relevant stage. But languages are also objects, and if they are also to be understood on an abstractionist conception, then it seems that the progression of languages also should be understood on a potentialist, dynamic abstraction conception. This paper explores the role of the progression of languages in dynamic abstraction.

Jon Litland (University of Texas at Austin) `Essentialist Potentialism'

Recently, potentialist approaches to mathematics (in particular, set theory) have received a lot of attention. The potentialist holds that, necessarily, for any objects, it is possible that there is a set of exactly those objects. (To avoid Russell's paradox the potentialist rejects that for any objects there is a set of them.) But what is the modality involved in these potentialist claims? The standard approach has been to take the modality to an interpretational modality, one that reinterprets the vocabulary. In this talk I briefly argue against the interpretational understanding of the modalities, and put forward an alternative: the modality is an essentialist modality, characterizable in a (higher-order) logic of essence. By using this essentialist modality we get a clearer statement of what it means for a set to be generated from its members. If time permits, I will consider what upshots this may have for the possibility of quantifying over absolutely everything.

Support

The workshop is supported by All Souls College and Lady Margaret Hall of the University of Oxford, and Norges Forskningsrådet (the Research Council of Norway) via the project Infinity and Intensionality: Towards a New Synthesis (no. 314435).

Conference organisers

Neil Barton, Øystein Linnebo, and Chris Scambler. For questions about the workshop, please contact Dr. Neil Barton (n.a.barton@ifikk.uio.no).

Engineering the Concept of Collection

Time and place: June 20, 2022–June 21, 2022, Georg Morgenstiernes hus and on Zoom

The history of mathematics and philosophy have seen many different concepts of collection: a set (understood as a gathering into one of previously available objects), a class (understood as defined by its membership condition, not by its members), a mereological sum, etc. Indeed, even a plurality (i.e. many objects) and a concept can be seen as a collection, since it makes sense to talk about their members (or instances).

Alongside these longstanding debates about the nature of collections, there are also questions of how exactly each conception should be made precise. Recent attempts to make sense of the ontology of combinatorial sets, for example, have proposed very different pictures of what they are like. This is especially clear in the debates on the nature of our thought concerning `the’ universe of sets: Does our concept of (combinatorial) set suffice to determine a unique and maximal universe, or does our concept and talk of sets admit of different multifarious interpretations?

These observations raise some general philosophical-mathematical questions. What concepts of collection do we have? Which, if any, of these concepts should we use? Or should we “(re-)engineer” one or more concepts of collection to produce concepts that are fit for purpose?

Further question may include:

• Should we use a single concept of collection or different concepts for different purposes? (Remember George Boolos: “I thought that set theory was supposed to be a theory about all, `absolutely' all, the collections that there are”.)
• Specifically, do we need both a combinatorial concept of set and a logical concept of class (cf. Parsons 1974, Maddy 1983, Linnebo 2006)?
• Should we strive for a single true theory based on one or more “correct” concepts of collection, or be pluralists and accept a multitude of equally legitimate, but often competing, theories?
• Can recent views on the nature of combinatorial sets (e.g. Hamkins, Linnebo, Woodin) be understood as instances of conceptual engineering?

Speakers

Carolin Antos (Universität Konstanz), Neil Barton (University of Oslo), Tim Button (University College London), Herman Cappelen (University of Hong Kong and University of Oslo), Laura Crosilla (University of Oslo), Kentaro Fujimoto (University of Bristol), Luca Incurvati (University of Amsterdam), Øystein Linnebo (University of Oslo), Stewart Shapiro (Ohio State University and University of Oslo)

The workshop will be hybrid (in-person and online). Registration is free but is required to attend the workshop. Please fill in this form to register.

Registration deadline: 19th of June 2022, 5 pm (CET). You will receive the Zoom invite by email in the evening of the 19th of June.

If you plan to attend in person in Oslo, please let the organisers know as soon as possible and in any case by the 10th of June.

Program

20th June, Georg Morgenstiernes hus, Room 652 -- (all times are in CET)

• 10:30-11:45 Herman Cappelen and Øystein Linnebo: Engineering the concept of collection: introductory remarks
• 11:45-13:00 Lunch
• 13:00-14:15 Luca Incurvati: Engineering the concept of set and engineering the concept of objectified property
• 14:15-14:30 Break
• 14:30-15:45 Kentaro Fujimoto: Plural, infinity, and impredicativity
• 15:45-16:00 Break
• 16:00-17:15 Laura Crosilla: On Weyl’s predicative concept of set

21st June, Georg Morgenstiernes hus, Room 452

• 10:30-11:45 Tim Button: MOON theory: Mathematical Objects with Ontological Neutrality
• 11:45-12:00 Break
• 12:00-13:15 Neil Barton: Engineering Set-Theoretic Concepts
• 13:15-14:15 Lunch
• 14:15-15:30 Carolin Antos: Engineering the concept of set in practice - a case for concept pluralism?
• 15:30-15:45 Break
• 15:45-17:00 Stewart Shapiro: Semantic indeterminacy, concept sharpening, set theories

Abstracts

Carolin Antos

Title: Engineering the concept of set in practice - a case for concept pluralism?

Abstract: Engineering the concept of set often involves meta-mathematical or philosophical discussions such as the foundational debate at the beginning of the 20th century or, more recently, the universe/multiverse debate in the philosophy of set theory. However one can also consider (at least part of) set-theoretic practice as a constant engineering process of the concept of set.

In this talk I would like to investigate if this practice gives rise to concept pluralism for the concept of set (as it arguably does for the concept of collection). To this end, I will connect the debate about conceptual engineering with the notion of concept pluralism that is discussed in philosophy of science. I argue that there are at least two theoretical uses the concept of set has, connected to Neil Barton's schematic and directed conception of set, and I will discuss this on the basis of the case of choichless large cardinals.

Neil Barton

Title: Engineering Set-Theoretic Concepts

Abstract: In this talk I'll present a main argument of a short book I'm working on entitled Engineering Set-Theoretic Concepts (I'm interested in comments on the draft, so please get in touch if you'd like to see it once ready). I'll first note that conceptual engineering has formed a part of set-theoretic activity since its inception as a mainstream area of mathematical research, and that the development of the iterative (and other) conceptions of set was in part responding to inconsistency in the naive set-concept. I'll then argue that whilst the iterative conception can be taken to be a consistent concept in its own right, it is deficient in various ways (in particular, it fails to tell us enough about the nature of infinite sets). Contemporary set theory, I'll argue, has now moved to a maximal iterative conception of set, and this conception is inconsistent. Many contemporary accounts of the ontology underlying set-theoretic practice should be conceived of as attempts to engineer consistent conceptions of the maximal iterative concept of set. I'll explain two such conceptions, the directed and schematic iterative conceptions of set. I'll tentatively conclude that discussion in the philosophy of set theory should focus less on the vexed and seemingly intractable issue of ontology, and instead concern itself more with the (nonetheless difficult) question of the relative theoretical virtues of alternative conceptions.

Tim Button

Title: MOON theory: Mathematical Objects with Ontological Neutrality.

Abstract: The iterative notion of set starts with a simple, coherent story, and yields a paradise of mathematical objects, which “provides a court of final appeal for questions of mathematical existence and proof”. But it does not present an attractive mathematical ontology: it seems daft to say that every mathematical object is “really” some (pure) set. My goal, in this paper, is to explain how we can inhabit the set-theorist’s paradise of mathematical objects whilst remaining ontologically neutral. I start by considering stories with this shape: (1) Gizmos are found in stages; every gizmo is found at some stage. (2) Each gizmo reifies (some fixed number of) relations (or functions) which are defined only over earlier-found gizmos. (3) Every gizmo has (exactly one) colour; same-coloured gizmos reify relations in the same way; samecoloured gizmos are identical iff they reify the same relations. Such a story can be told about (iterative) sets: they are monochromatic gizmos which reify one-place properties. But we can also tell such stories about gizmos other than sets. By tidying up the general idea of such stories, I arrive at the notion of a MOON theory (for Mathematical Objects with Ontological Neutrality). With weak assumptions, I obtain a metatheorem: all MOON theories are synonymous. Consequently, they are (all) synonymous with a theory which articulates the iterative notion of set (LT+). So: all MOON theories (can) deliver the set theorist’s paradise of mathematical objects. But, since different MOON theories have different (apparent) ontologies, we attain ontological neutrality. My metatheorem generalizes some of my work on Level Theory. It also delivers a partial realization of Conway’s “Mathematician’s Liberation Movement”.

Herman Cappelen and Øystein Linnebo

Title: Engineering the concept of collection: introductory remarks

Abstract: The history of mathematics and philosophy have seen many different concepts of collection: a combinatorial set (understood as a gathering into one of previously available objects), a logical class (understood as defined by its membership condition, not by its members), a mereological sum, etc. My introductory remarks discuss some questions that this observation raises. What concepts of collection do we have? Which, if any, of these concepts should we use? Or should we “(re-)engineer” one or more concepts of collection to produce concepts that are more fit for purpose? I explain how we appear to need two importantly different concepts of collection: combinatorial sets (as on the famous iterative conception) and logical classes (cf. Parsons 1974, Maddy 1983, Linnebo 2006). The former is highly natural and well understood. Although the latter is less well understood, I argue that it too admits of a highly natural development.

Laura Crosilla

Title: On Weyl’s predicative concept of set

Abstract: A key component of the early 20th century literature on predicativity are variants of the traditional logical concept of set, according to which a set is the extension of a concept. In the first chapter of Das Kontinuum (1918), Hermann Weyl carefully spells out a new step-by-step construction of sets as extensions of predicative properties of the natural numbers, with the purpose of providing a strong and lasting foundation for mathematical analysis (and beyond). Weyl clearly contrasts his own concept of set with a quasi-combinatorial concept of set, which he finds wanting when working with infinite sets. It is remarkable that in Das Kontinuum Weyl deliberately moves away from a set-theoretic foundation of mathematics, which was instead the context of his Habilitation lecture of 1910. Weyl’s predicative concept of set and his predicative analysis have had lasting impact on mathematical logic and constructive mathematics. Arguably, a variant of a predicative concept of set as extension of a predicative property is also at the heart of some contemporary constructive type theories, which are increasingly popular due to their computational applications. In this talk, I argue that predicativism and Weyl’s changing positions on the foundations of mathematics are significant examples that are bound to enrich the contemporary debate on concept engineering in mathematics.

Kentaro Fujimoto

Title: Plural, infinity, and impredicativity.

Abstract: Plural logic receives increasing attention and popularity. It counts plural terms among the irreducible basic vocabulary of logic that are of a distinct kind from singular terms, and allows quantification into plural term position. Advocates of plural logic often appeal to two benefits: namely, plural logic adequately formalizes plural discourse in English and is ontologically innocent. In this talk, I examine the legitimacy and necessity of plural logic and compare it with several alternative theories from philosophical, linguistic, and mathematical perspectives. Then I will argue that the two alleged benefits of plural logic do not necessarily endorse it. This is a work in progress.

Luca Incurvati

Title: Engineering the concept of set and engineering the concept of objectified property

Abstract: I will start by clarifying the extent to which the defence of the iterative conception in Incurvati 2020 is compatible with pluralism. The idea is that some conceptions are better than others for certain goals. The approach is functionalist about concepts: sharpenings of concepts are selected on the basis of how well they fullfill a certain function. I will then move on to discuss the case of objectified properties. I will again start from a functionalist approach and take as a starting point the expressive function of objectified-property talk. I will then report work in progress on an inferential deflationist theory of objectified properties, which is motivated by the functionalist approach and takes the meaning of objectified-property talk to be completely explained in terms of its inferential relation to property talk. I will discuss formal and philosophical aspects of the theory, in particular its relationship with Maddy's work on classes and the question of realism about objectified properties.

Stewart Shapiro

Title: Semantic indeterminacy, concept sharpening, set theories

Abstract: Friedrich Waismann once suggested that mathematical concepts are not subject to open-texture; they are “closed”. In other work, I have highlighted some traditional mathematical notions that were, at one time, open-textured. One of them is the notion of “polyhedron” following the history sketched in Lakatos’s Proofs and Refutations. Another case is computability, which has now been sharpened into a plausibly closed notion. There are also some mathematical notions that have longstanding, intuitive principles underlying them, principles that later proved to be inconsistent with each other, sometimes when the notion is applied to cases not considered previously (in which case it it perhaps an instance of open-texture). One examples is “same size”, which is or was governed by the part whole principle (one of Euclid’s Common Notions) and the one-one principle, now called “Hume’s Principle”. Another is the notion of continuity. The purpose of this talk is to explore the notion of “set” and other related (perhaps once identical) notions like class, totality, and the like. We tentatively put forward the thesis that this notion, too, is or was subject to open-texture (or something like it) and could be (and has been) sharpened in various ways. This raises some questions concerning what the purposes of a (sharpened) theory of sets are to be. And, in that context, the role of trying to give non-ad-hoc explanations or answers to various questions.

The workshop is organized within the Infinity and Intensionality project, jointly with ConceptLab.

Organizers

Neil Barton, Laura Crosilla and Øystein Linnebo

Weyl 2021: A workshop dedicated to Hermann Weyl's Philosophy of Mathematics

Time and place: Dec. 8, 2021–Dec. 10, 2021, Georg Morgenstiernes hus, room 652 and on Zoom

Hermann Weyl (1885-1955) was perhaps one of the last mathematicians to produce profound, influential work across a vast territory in mathematics, physics and philosophy. He proposed a highly original, radically new foundation of analysis in his 1918 book “Das Kontinuum”. In 1921, Weyl published “Über die neue Grundlagenkrise der Mathematik”, where he declared:

“So I now abandon my own attempt and join Brouwer.”

To celebrate the 100 years since the publication of Hermann Weyl’s “Über die neue Grundlagenkrise der Mathematik”, we will hold a Workshop dedicated to Hermann Weyl’s Philosophy of Mathematics, with special (but not exclusive) focus on his intuitionistic turn.

Speakers

• Stefania Centrone (TU Berlin)
• Laura Crosilla (University of Oslo)
• José Ferreirós (Universidad de Sevilla)
• Janet Folina (Macalester College),
• Dagfinn Føllesdal (University of Oslo)
• Mirja Hartimo (University of Helsinki)
• Øystein Linnebo (University of Oslo)
• Henri Lombardi (Université de Franche-Comté)
• Pierluigi Minari (Università degli Studi di Firenze)
• Stefan Neuwirth (Université de Franche-Comté)
• Michael Rathjen (University of Leeds)
• Wilfried Sieg (Carnegie Mellon University)
• Iulian Toader (Universität Wien)
• Mark van Atten (Husserl Archive (CNRS / ENS), Paris)

The workshop will be online. Registration is required to attend the workshop. Please fill in this form to register.

Registration deadline: 6 December 12pm (CET). You will receive the Zoom invite on 7 of December by email. We encourage you to keep your webcam on whenever possible to help recreate the atmosphere of an in-person meeting.

The workshop is organized within the Infinity and Intensionality project, jointly with ConceptLab.

Schedule (all times are CET)

Wednesday 8 of December

• 14:30-15:00: Dagfinn Føllesdal, Opening Lecture
• 15:10-16:10: Stefania Centrone and Pierluigi Minari, Oskar Becker's "On the logic of modalities": between modal logic and philosophy of mathematics
• 16:20-17:20: Wilfried Sieg, Mathematical structuralism: Weyl’s positions
• 17:30-18:30: Laura Crosilla and Øystein Linnebo, Weyl and two kinds of potential domains

Thursday 9 of December

• 14:30-15:30: Michael Rathjen, On “real” statements
• 15:40-16:40: José Ferreirós, Second thoughts on set theory
• 16:50-17:50: Iulian Toader, Normative Quantifiers
• 18:00-19:00: Janet Folina, Generality, Circularity and Foundations

Friday 10 December

• 14:30-15:30: Mirja Hartimo, From Husserl's definiteness to Weyl’s Über die neue Grundlagenkrise der Mathematik, 1921
• 15:40-16:40: Mark van Atten, Intuitionistic induction
• 16:50-17:50: Henri Lombardi and Stefan Neuwirth, Hermann Weyl and Paul Lorenzen on predicative mathematics
• 17:55-18:50: General Discussion

Titles and Abstracts (in alphabetical order)

Stefania Centrone and Pierluigi Minari

Oskar Becker's "On the logic of modalities": between modal logic and philosophy of mathematics

In his 1930 Essay "On the Logic of Modalities" O. Becker pursued two loosely related goals. The first one, more technical in nature, was to find axiomatic conditions capable to reduce to the finite the number of irreducible modalities in Lewis’s Survey system. The second one, more philosophically oriented and, in a sense, much more ambitious, was to treat the logic of modalities from a phenomenological perspective and to understand, from this perspective, the philosophical and logical-ontological problems underlying, and posed by, Intuitionism. In our talk, we will focus on Becker's discussion of Weyl's position in The Continuum and on Becker's anticipation of Goedel's modal translation of intuitionistic logic.

Laura Crosilla and Øystein Linnebo

Weyl and two kinds of potential domains

According to Weyl, “inexhaustibility” 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. In this talk, we present Weyl's distinction and explain its logical and philosophical significance. In particular, the distinction sheds light on the contemporary debate about potentialism, which in turn affords a deeper understanding of Weyl.

José Ferreirós

Second thoughts on set theory

The talk will describe some of the historical circumstances surrounding Weyl’s foundational contributions. Concerning Weyl it will deal primarily with the period before 1921, both in Göttingen and then Zurich, where Weyl was surrounded by people like Polya, Zermelo, Bernays, etc. In Das Kontinuum Weyl described the way in which he gradually found his way out of the circle of set-theoretic ideas -- reading his 1910 Habilitation lecture, one can indeed confirm that Weyl was still at that time a rather classical follower of set-theoretic mathematics along the lines of Dedekind, Cantor and Hilbert. The pattern of an initial enthusiasm for set theory, followed by subsequent second thoughts, is not at all unusual. I will discuss how it is also present in other experts of the first half of the 20th century, such as Borel, Lebesgue, Lusin, lesser known figures like Rey Pastor, or even van der Waerden.

Janet Folina

Generality, Circularity and Foundations

Weyl and Poincaré both argued that there was something circular, and thus objectionable, in logicist and set-theoretic foundationalism. Critics have dismissed this type of objection as presupposing constructivism - or worse, as psychologistic. With help from some views from Bernays I aim to show that the objections are substantive, and reveal an important disagreement about what we are looking for in a foundation for mathematics.

Bernays distinguishes two concepts of generality, and argues that neither subsumes the other. After fleshing out this view I’ll argue that it clarifies and supports the circularity objections. Finally, I’ll propose that the different types of generality are aligned with different approaches to the foundations of mathematics: the logical and the (more broadly) epistemological. Bernays’ distinction thus provides another way to understand Weyl’s turn to Brouwer and intuitionism in 1921.

Mirja Hartimo

From Husserl's definiteness to Weyl’s Über die neue Grundlagenkrise der Mathematik, 1921

Since the turn of the century, Husserl’s philosophy of mathematics combined structuralist and constructivist elements. Thus, in his Doppelvortrag (1901) Husserl complements his merely formal notion of definiteness as categoricity with a more concrete and constructive notion of definiteness, definiteness in “pregnant sense.” In this sense, definiteness requires establishing the existence of the domain of the theory by so-called “existential axioms”. In Ideas I (1913), responding to the debates about the paradoxes, Husserl attempted to do this by “regional axioms,” with which he sought to capture the essences that the objects have apriori, but synthetically. In this talk, after having noted the similarities between Husserl’s and Weyl’s notion of “extensional definiteness” in Das Kontinuum, I will briefly examine, from a Husserlian point of view, the further development of the notion in Weyl’s Brouwerian 1921 paper Über die neue Grundlagenkrise der Mathematik.

Michael Rathjen

On “real” statements

As late as 1954, referring to Hilbert's program, Weyl writes that ``Intuitive reasoning is required and used merely for establishing the consistency of the game''. The consistency of a theory T (assumed to contain a modicum of arithmetic) established in this way, however, entails that a certain class of theorems of T has to be accepted as true even by the finitist and a fortiori by the intuitionist, as Kolmogorov pointed out already in 1925.

These formulas are of universal form, i.e., they start with a string of universal quantifiers followed by a quantifier free matrix and are usually referred to as the Pi1 formulas.

They were assigned a privileged status by Hilbert who termed them real statements. The class of Pi1 formulas has some famous denizens, notably Fermat's last theorem, the Riemann hypothesis and Goldbach's conjecture. There is still no satisfactory answer to Kreisel's question ``What more than its truth do we know if we have proved a Pi1 theorem in a theory T?''.

For the larger collection of Pi2 theorems, i.e., those starting with a universal quantifier followed by an existential quantifier, there is an answer in terms of growth rates of functions.

In this talk, I plan to discuss possibilities of classifying Pi1 theorems by unearthing their transfinite content.

Wilfried Sieg

Mathematical structuralism: Weyl’s positions

The “structuralism” of modern mathematics found one of its earliest and clearest expressions in Dedekind’s work, in particular, in his foundational essay Was sind und was sollen die Zahlen? [1]. Dedekind rejected (Kantian) intuitions for the grounding, but also the systematic development of mathematics; that is clear from the introduction to his essay and the contemporaneous letter to Keferstein [2]. Weyl was critical of Dedekind’s as well as Hilbert’s perspective on “structural axiomatics” and the associated “abstract conception of proof”. That criticism emerged also in Weyl’s discussions with Emmy Noether in the early 1930s; see [3]. However, in one of his last papers [4], Weyl sketched a more balanced view of the axiomatic and constructive approaches.

[1] Dedekind, R. 1888, Was sind und was sollen die Zahlen?, Vieweg. Braunschweig.

[2] Dedekind, R. 1890, Letter to Keferstein, in: van Heijenoort (ed), From Frege to Gödel, Harvard University Press, Cambridge, 1967, pp. 98-103.

[3] Toader, I.D. 2021, Why did Weyl think that Emmy Noether made algebra the Eldorado Of Axiomatics?, HOPOS 11 (1), pp. 122-142.

[4] Weyl, H. 1985, Axiomatic versus constructive procedures in mathematics (manuscript for a talk around 1953), edited by T. Tonietti, The Mathematical Intelligencer 7 (4), pp. 10-17 and 38.

Iulian Toader

Normative Quantifiers

Weyl’s reasons for his move towards intuitionism, in particular his ideas on quantification, were quite novel and really interesting. After presenting some likely historical sources, I argue that he believed the law of excluded middle is meaningless for infinitary mathematics not because quantified statements, themselves, are meaningless, but because they are rules for proper judgment. Infinitary mathematics, according to Weyl, is a normative system.

(Dedicated to the memory of David Charles McCarty)

Mark van Atten

Intuitionistic induction

In the intuitionistic tradition after Brouwer, one finds two different answers to the question what the evidence of the principle of induction consists in. In this talk, both will be presented in their historical context, and compared with a reconstruction of Brouwer’s view. Finally, Brouwer’s view thus reconstructed will be juxtaposed with Weyl’s.

Organisers

Laura Crosilla, Øystein Linnebo and Michael Rathjen

This workshop is organized jointly with ConceptLab and the Infinity and Intensionality project, funded by the Norwegian Research Council (NFR)

Countabilism Workshop

Countabilism is the view according to which every infinite collection is countable. We discuss this view in an online workshop.

Time and place: Nov. 10, 2021 3:00 PM–6:30 PM, Zoom

Speakers

Jessica Wilson (University of Toronto), David Builes (Princeton University), Stewart Shapiro (Ohio State University), Chris Scambler (Oxford University), Øystein Linnebo (University of Oslo), Laura Crosilla (University of Oslo).

Schedule (all times are CET)

• 15:00 -- 15:50: Chris Scambler, Two Paths to Countabilism
• 16:00 -- 16:50: Jessica Wilson and David Builes, In Defense of Countabilism
• 17:00 -- 17:50: Laura Crosilla, Øystein Linnebo and Stewart Shapiro, Predicativity and Countabilism
• 18:00 -- 18:30: General discussion

The workshop will be held on Zoom. You need to register to attend the workshop.

Registration deadline: 8 November 2021. Please fill in this form to register.

Titles and Abstracts

Jessica Wilson and David Builes

Title: In Defense of Countabilism

Abstract: Inspired by Cantor's Theorem (CT), orthodoxy takes infinities to come in different sizes. The orthodox view has had enormous influence in mathematics, philosophy, and science. We will defend the contrary view---Countablism---according to which, necessarily, every infinite collection (set or plurality) is countable. We first argue that the potentialist or modal strategy for treating Russell's Paradox, first proposed by Parsons 2000 and developed by Linnebo 2010, 2013 and Linnebo and Shapiro 2019, should also be applied to CT, in a way that vindicates Countabilism. Our discussion dovetails with recent independently developed treatments of CT in Meadows 2015, Scambler 2021, and Pruss 2020, aimed at establishing the mathematical viability, and therefore epistemic possibility, of Countabilism. Unlike these authors, our goal isn't to vindicate the mathematical underpinnings of Countabilism. Rather, we aim to argue that, given that Countabilism is mathematically viable, Countabilism should moreover be regarded as true. After clarifying the modal content of Countabilism, we canvas certain of Countabilism's many positive implications, including that Countabilism provides the best account of the pervasive independence phenomena in set theory, and that Countabilism has the power to defuse several persistent puzzles and paradoxes found in physics and metaphysics. We conclude that the theoretical advantages of Countabilism far outweigh its potential downsides.

Chris Scambler

Title: Two Paths to Countabilism

Abstract: Countabilism -- the idea that, Cantor's theorem notwithstanding, all sets are countable -- has a long history in philosophy of mathematics, and there are many ways the doctrine has been supported. In this presentation I will sketch two such paths of support of recent heritage -- one with roots in forcing potentialism, and the other with roots in the theory of the real numbers -- and will try to show that each can be used to strengthen the case made by the other.

Laura Crosilla, Øystein Linnebo and Stewart Shapiro

Title: Predicativity and Countabilism

Abstract: In his classic “Systems of predicative analysis” (Feferman 1964), Solomon Feferman highlights an important aspect of predicativism, namely the view that some totalities are inherently potential. A direct connection between predicativism and potentialism is also key to Poincaré’s later work on predicativity. Building on the modal analysis of potentialism in (Linnebo and Shapiro 2018), we present an analysis of predicativism as a form of potentialism, clarify the predicativist interpretation of Cantor’s theorem, and discuss whether predicativism is a form of countabilism.

Organisers

Neil Barton, Laura Crosilla and Øystein Linnebo