Michael Hallett (McGill)
Justin Clarke-Doane (New York University)
“Multiple Reductions Revisited: Property Theory in Mathematics”
In this paper I develop two relatively unexplored lines of objections to Paul Benacerraf’s famous argument from multiple reductions. Instead of arguing that Benacerraf failed to establish that numbers couldn’t be irreducible objects of some sort, I argue that he failed even to establish that numbers couldn’t be reduced. In particular, I argue that the conditions which Benacerraf claims that a correct reduction of the numbers must satisfy are far too leniant, and that the actual such conditions a plausibly satisfied uniquely by a reduction of numbers to properties of collections. Not only did Benacerraf seriously underestimate the extent of the data that is relevant to the identity of the numbers, but he also failed to see that a correct reduction of them must account for that data best.
Sam Cowling (University of Manitoba)
“Hypergunk and Inaccessible Fictionalism” (not being presented)
In this paper, I discuss a conflict between the commitments of mereology and set theory and develop a strategy for resolving it. In section one, I show that two attractive theses, one modal and mereological and one set-theoretic, entail a contradiction. In sections two and three, I outline a position that takes mereology seriously and rejects the troublesome set-theoretic thesis. In doing so, I show how one can plausibly defend the perfect generality of mereology along with the possibility of hypergunk at the expense of adopting a fictionalist account of certain set-theoretic posits (i.e., strongly inaccessible cardinals).
Tracy Lupher (University of Texas at Austin)
“How to Construct Unitarily Inequivalent Representations in Quantum Field Theory”
A topic of philosophical interest in the foundations of quantum field theory literature is the physical significance of unitarily inequivalent representations. That literature is formidable; it presupposes familiarity with the algebraic framework. A key innovation here is to generate such representations in an accessible manner using a simple model that serves to construct a broad spectrum of such representations in the physics literature. Aside from that unifying feature, other novel results are obtained including a new version of Haag’s theorem, a clear explanation of the limitations of the Fock representation, and the under-determination of the vacuum state.
John Manchak (University of California at Irvine) Clifton Memorial Book Prize
“Observational Indistinguishability and Geodesic Incompleteness”
It has been suggested by Clarke Glymour that the spatio-temporal structure of the universe might be underdetermined by all observational data that could ever, theoretically, be gathered. Becuase the plight of the cosmologist seemed to be so discouraging in this regard, David Malament considered the relationship of between global properties and OI spacetimes. This information is helpful to the cosmologist. It allows, in principle, one to reject some spacetime models based on observational evidence. In this paper, I consider a property that Malament did not: geodesic incompleteness. In the light of the findings, it seems that (for the most part) the predicament of the cosmologist it not good. Quite generally, versions of geodesic incompleteness are not conserved even under the strongest formulations of OI. A possible objection is then considered: In some of my examples, compact sets of points are excised from Minkowski spacetime. John Earman has conjectured that such examples are not physically reasonable (they don’t satisfy the right standard energy conditions). I offer a proof that Earman’s conjecture is false.
Aidan McGlynn (University of Texas at Austin)
“Iterations and Limitations”
The iterative conception of set has been defended as a natural and non-arbitary successor to the inconsistent native conception, but in ‘The Iterative Conception of Set’ George Boolos showed that the hierarchical picture of the set theoretic universe given to us by this conception of set fails to lend support to some of the axioms of ZFC, most notably choice and replacement. Both these axioms are delivered by a rival conception of set–the limitation of size conception–but unhappily this puts the axioms of power set and infinity beyond our reach, and has struck many as merely a technical device designed to avoid the paradoxes, rather than a genuine elcidation of our conception of set. Boolos has suggested that perhaps our conception of set is a hybrid of the leading thoughts behind the iterative conception and limitation of size, and in this paper I begin an assesment of the prospects of such a conception. I argue that even if this hybrid conception–the limiatation of iteration conception, as I call it–can deliver all the axioms of ZFC, it does so only if we are willing to make assumptions justified–if at all–only on pragmatic grounds. Insofar as out project is that of providing conceptual grounds on which to believe the axioms, I conclude that with the limitation of iteration conception we take on step forward, but two steps back.
Bryan Pickel (University of Texas at Austin)
“On the (Lack of) Implications of Mathematical Syntax for Mathematical Ontology”
I explore two standard syntactic analyses of the English number words: as adjectives and as singular terms. I discuss but take no stand on the syntactic priority thesis that holds that one’s grasp of ontological categories is determined by one’s grasp of syntactic categories. One who accepts the principle will be inclined to hold that if number words are syntactically singular terms, then the numbers themselves must be assayed Platonistically. I argue that there is no conclusive reason for analyzing the number words as singular terms. I provide some further evidence suggesting that they do not fit the syntactic profile of singular terms. I conclude that even accepting the syntactic priority thesis cannot force the nominalist into Platonism.
Florian Steinberger (Cambridge University)
“On Multiple-Succedent Sequent Calculi”
Anti-realists have introduced the notion of proof theoretic arguments for the adoption of intuitionistic or other constructive systems of logic. The central idea is to derive certain constraints on the possible form of the inference rules governing the logical constants departing from general considerations concerning the properties of an adequate theory of meaning. Based on a case study of the calculus of sequents, I show that proof-theoretic arguments hinge on the particular, largely arbitrary structural features of natural deduction systems and therefore fail to achieve their objective.
Darren McDonald, Robert Moir, Jenny Noland