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).