Structure, Identity and Abstraction workshop

Structure and Identity and Abstraction workshop

Organizers: Leon Horsten and Philip Welch

Sponsored by: Institute for Advanced Studies Interdisciplinary Research Grant and an AHRC Research Grant for The Foundations of Structuralism

Venue details

Burwalls

The venue for the workshop is the music room at Burwalls. The address is Bridge Rd, Leigh Woods, Bristol, Avon BS8 3PD


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Programme

See below for abstracts.

Thursday,10 December: Structure and identity

Time Event
9.00 - 9.30 Welcome & coffee
9.30 - 10.00 Opening: James Ladyman (Bristol)
10.30-11.30 Coffee
11.30.12.30 Wolfgang Spohn (Konstanz)Abstract
12.30-2.30  Lunch
14.30-15.30 Jeff Ketland (Edinburgh)Abstract
15.30-16.30 Christopher Von Bülow (Konstanz) Abstract
16.30-17.00 Coffee
17.00-18.00 Adam Caulton (Cambridge) Abstract

Friday, 11 December: Identity and abstraction

Time Event
9.30 - 10.30 F.A. Muller (Rotterdam and Utrecht) Abstract
10.30-11.00 Coffee
11.00-12.00 Øystein Linnebo (Bristol) Abstract
12.00-14.30 Lunch
14.30-15.30 Gabriel Uzquiano (Oxford) Abstract
15.30-16.30 Rafal Urbaniak (Ghent & Gdansk)
16.30-17.00 Coffee
17.00-18.00 Roy Cook (Minnesota) Abstract

Abstracts

Christopher von Bülow

Places in Structures, Places of Relations, and Mechanisms for Recognition

Stewart Shapiro explicates mathematical objects as 'places in structures', where structures (e.g., the natural-number structure) are both properties of systems (e.g., the natural numbers with successor) and systems having these properties. The places in a given structure-as-system (e.g., the numbers) constitute that system's domain and have 'relational essences': their nature consists in the relations they have to each other by virtue of belonging to the system (e.g., who is successor of whom); they do not have unwanted surplus properties (e.g., numbers having elements). Furthermore, these places are somewhat like offices in being 'occupiable' by the objects (e.g., certain sets) of systems having the structure-as-property. One challenge for Shapiro's account is the indiscernibility problem: for some structures (e.g., Euclidean space, the field of complex numbers) there are distinct places (e.g., i and -i) which have exactly the same relations to the rest of the structure-as-system; if these relations constitute their essences then they are indiscernible and thus, it seems, identical. I distinguish 'relational-essence places' and 'office places'. I don't believe in the former; but the latter (though bad candidates for mathematical objects) are also threatened by the indiscernibility problem. I try to defuse the problem by assimilating the office places of structures to the argument places of relations. For symmetrical relations, the same problem occurs: their argument places are indiscernible. To get a grip on the indiscernibility problem for argument places, I make a proposal for how to better understand relations: via mechanisms (not necessarily effective ones) for recognizing instantiations of these relations. Their argument places then are abstractions from the 'information channels' such mechanisms must incorporate to import the utilized information about the objects to be compared. Argument places are abstract with a vengeance. Can my proposal save them from the indiscernibility problem?

Adam Caulton

Identity, indiscernibility and individuality: logical results and metaphysical options

In this talk, which is a report of research carried out with Dr Jeremy Butterfield, I survey some issues of identity, indiscernibility and individuality in a rather general setting (namely, first-order languages), and present four salient and rival metaphysical positions which emerge from these considerations.  I see these four positions as arising from the different answers one may give to two questions, each of which is based on an important distinction.  The first distinction is between reductive and non-reductive accounts of identity, and thus pertains to the truth or falsity of (some version of) the principle of the identity of indiscernibles.  The second distinction, perhaps less emphasised, is between identity and individuality.  Significantly, these distinctions gives logical space to a position which takes a reductive approach to individuality but a non-reductive approach to identity.  I will explore the semantics suitable for such a position, along with the other three metaphysical positions, bringing in some relevant considerations from physics.  Along the way, I will also talk about the connections between two kinds of indiscernibility relevant to the Hilbert-Bernays-Quine reductive account of identity and the existence of symmetries (a.k.a. automorphisms) on the domain of discourse.

Roy Cook

"In Favor of Strong Stability"

One of the most pressing questions facing the defenders of neo-logicism is responding to the Bad Company objection - that is, providing a philosophically well-motivated account that separates the 'good' abstraction principles (e.g. Hume's Principle) from the 'bad' one (e.g. Basic Law V). A number of different formal criteria have been proposed as means for drawing this line. I shall begin by briefly surveying these different criteria, answering a few open technical questions and identifying those remaining questions that lack answers. An argument will then be presented that the strictest of these - strong stability - is the correct criterion for acceptability of abstraction principles. I shall conclude by briefly defending strong stability from three extant objections.

Jeffrey Ketland

Identity and Indiscernibility

The notion of strict identity is sometimes given an explicit second-order definition: objects with all the same properties are identical. Here, a somewhat different problem is raised: Under what conditions is the identity relation on the domain of a structure first-order definable? A structure may have objects that are distinct, but indiscernible by the strongest means of discerning them given the language (the indiscernibility formula). Here a number of results concerning the indiscernibility formula, and the definability of identity, are collected and a number of applications discussed.

Øystein Linnebo

Reference by abstraction

I describe a language that appears to allow reference to directions and other kinds of abstract objects. Some semantic theories take this appearance at face value, whereas others explain it away and eschew any mention of abstract objects. I develop some arguments in favour of the former sort of semantics. This amounts to a form of reference by abstraction. I end by discussing a respect in which the use of a language is essentially prior to semantic theorizing about the language.

F A Muller

Deep Ontology

Since physics is our most general science and metaphysics aims for generality about reality, the institutional fact that philosophy of physics and metaphysics have their own, separate domains of discourse is odd. Metaphysics without philosophy of physics is shallow and philosophy of physics without metaphysics is superficial. Hence Deep Ontology is the proposed name for a philosophical discourse on the nature of reality in which both philosophers of physics and metaphysicians prominently take part, and in which they hopefully interact fruitfully. (A variety of papers and books can already be seen as belonging to this discourse.)

We attempt to give an example of a piece of work that qualifies as belonging to Deep Ontology. We begin by criticising an elaboration of an argument in Mind, due to K. Hawley (2009), to the effect that an appeal to relations cannot defend Leibniz's Principle of the Identity of Indiscernibles (PII) when faced with putative counter-examples. Next we argue that in the un-elaborated argument, which goes back to R. Barcan Marcus (1993) and has recently been expounded by S. French & D. Krause (2006), the fallacy of equivocation is committed. Then we present two arguments in favour of the so-called Discerning Defence over the so-called Summing Defence of PII.

A crucial element in some of our arguments is the recently discovered discernibility of all elementary particles when described quantum-mechanically, fermions as well as bosons included. We claim that a new metaphysical category of objects has been discovered in physical reality: relationals.

A genuine metaphysical discovery in the very first self-conscious contribution to Deep Ontology: this should make our minds drool….

Wolfgang Spohn

How are mathematical objects constituted? A structuralist suggestion

The talk proposes to amend structuralism in mathematics by saying what places in a structure and thus mathematical objects are. They are the objects of the canonical system realizing a categorical structure, where that canonical system is a minimal system in a specific essentialistic sense. It would thus be a basic ontological axiom that such a canonical system always exists. This way of conceiving mathematical objects is underscored by a defense of an essentialistic version of Leibniz' principle according to which each object is uniquely characterized by its proper and possibly relational essence (where "proper" means "not referring to identity").

Ineffability and abstraction (with Stewart Shapiro)

Gabriel Uzquiano

In this paper, we look at the prospects of a neo-Fregean foundation for set theory. We present a family of theories of extensions based on the combination of a restricted version of Basic Law V and a reasonably pure principle of reflection. Each of the theories we study can be motivated by the inchoate thought that the universe of extensions is ineffable and some of them can be captured by suitable abstraction principles for extensions. However, their reformulation into abstraction principles comes at a price