Categoricity

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those involving the distinction between characterizing a system and axiomatizing the truths of a syste

Potential infinity, abstraction principles and arithmetic (Leniewski Style)

This paper starts with an explanation of how the logicist research program can be approached within the framework of Leśniewski’s systems. One nice feature of the system is that Hume’s Principle is derivable in it from an explicit definition of natural numbers. I generalize this result to show that all predicative abstraction principles corresponding to second-level relations, which are provably equivalence relations, are provable. However, the system fails, despite being much neater than the construction of Principia Mathematica (PM). One of the key reasons is that, just as in the case of the system of PM, without the assumption that infinitely many objects exist, (renderings of) most of the standard axioms of Peano Arithmetic are not derivable in the system. I prove that introducing modal quantifiers meant to capture the intuitions behind potential infinity results in the (renderings of) axioms of Peano Arithmetic (PA) being valid in all relational models (i.e. Kripke-style models, to be defined later on) of the extended language. The second, historical part of the paper contains a user-friendly description of Leśniewski’s own arithmetic and a brief investigation into its properties

Stable frames and weights

Was paper 839 in the author's list until winter 2023 when it was divided into three. Part I: We would like to generalize imaginary elements, weight of ortp$(a,M,N), {\mathbf P}$-weight, ${\mathbf P}$-simple types, etc. from [She90, Ch. III,V,\S4] to the context of good frames. This requires allowing the vocabulary to have predicates and function symbols of infinite arity, but it seemed that we do not suffer any real loss. Part II: Good frames were suggested in [She09d] as the (bare bones) right parallel among a.e.c. to superstable (among elementary classes). Here we consider $(\mu, \lambda, \kappa)$-frames as candidates for being the right parallel to the class of $|T|^+$-saturated models of a stable theory (among elementary classes). A loss as compared to the superstable case is that going up by induction on cardinals is problematic (for cardinals of small cofinality). But this arises only when we try to lift. For this context we investigate the dimension. Part III: In the context of Part II, we consider the main gap problem for the parallel of somewhat saturated model; showing we are not worse than in the first order case

Logical Conventionalism

Once upon a time, logical conventionalism was the most popular philosophical theory of logic. It was heavily favored by empiricists, logical positivists, and naturalists. According to logical conventionalism, linguistic conventions explain logical truth, validity, and modality. And conventions themselves are merely syntactic rules of language use, including inference rules. Logical conventionalism promised to eliminate mystery from the philosophy of logic by showing that both the metaphysics and epistemology of logic fit into a scientific picture of reality. For naturalists of all stripes, this was paradise. Alas, paradise was lost. By the end of the twentieth century, logical conventionalism had been almost universally abandoned. But more recently, it has been revitalized. This chapter provides an overview of logical conventionalism and its history, clears away misunderstandings, and briefly responds to both the canonical objections to conventionalism (from Quine, Dummett, Boghossian, and many others) as well as new objections (from Field and Golan). From paradise lost, to paradise regained: logical conventionalism is back

Carnap and the Ontology of Mathematics

In this thesis I investigate Rudolf Carnap's philosophy of mathematics. Most philosophers assume that the nature of mathematics raises deep philosophical questions, which call for theories about how we manage to know about and interact with abstract objects. Carnap's position, in contrast, is deflationary: he aims to show that we can take mathematics at face value without having to answer questions about the metaphysical status of mathematical objects. If Carnap is right, there is thus no need for a philosophy of mathematics as it is usually understood at all. The main argument of my thesis is that Carnap's position is unstable, since his own commitments force him to make at least some ontological assumptions about syntax, i.e. entities such as letters, strings, and proofs. My claim that Carnap needs to accept some ontological questions as being in good shape goes against the received view in the secondary literature. Since the late 1980s interest in Carnap's philosophy of mathematics has been growing, mostly in the wake of important papers by Michael Friedman, Warren Goldfarb, and Thomas Ricketts. These and other scholars have forcefully defended Carnap against objections by, among others, Kurt Gödel, W. V. Quine, and Hilary Putnam. The most powerful challenge to Carnap's view, however, can actually be found in a less well-known paper by the logician E. W. Beth. The core of my thesis is thus a new interpretation of what I call Beth's argument from non-standard models, which relies on Gödel's incompleteness theorems and targets Carnap's claim that mathematics is analytic. I show that my reconstruction of Beth's argument is more charitable to the text than competing interpretations in the secondary literature, and argue that it is also more powerful since extant defences of Carnap cannot be applied.Funding was provided by the Arts and Humanities Research Council, the Cambridge Trust, the Aristotelian Society, the Royal Institute of Philosophy, and the German-American Fulbright Commission

Symmetries and Paraparticles as a Motivation for Structuralism

This paper develops an analogy proposed by Stachel between general relativity (GR) and quantum mechanics (QM) as regards permutation invariance. Our main idea is to overcome Pooley's criticism of the analogy by appeal to paraparticles. In GR the equations are (the solution space is) invariant under diffeomorphisms permuting spacetime points. Similarly, in QM the equations are invariant under particle permutations. Stachel argued that this feature--a theory's `not caring which point, or particle, is which'--supported a structuralist ontology. Pooley criticizes this analogy: in QM the (anti-)symmetrization of fermions and bosons implies that each individual state (solution) is fixed by each permutation, while in GR a diffeomorphism yields in general a distinct, albeit isomorphic, solution. We define various versions of structuralism, and go on to formulate Stachel's and Pooley's positions, admittedly in our own terms. We then reply to Pooley. Though he is right about fermions and bosons, QM equally allows more general types of symmetry, in which states (vectors, rays or density operators) are not fixed by all permutations (called `paraparticle states'). Thus Stachel's analogy is revived.Comment: 45 pages, Latex, 3 Figures; forthcoming in British Journal for the Philosophy of Scienc