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