Title: Ontology Without Tears: A Solution to the Problem of Abstract Objects (That Even a Naturalist Could Love)
Abstract: Some philosophers have suspicions about the analyses of mental states and linguistic expressions which employ abstract objects, or properties and propositions, or possible objects. These entities seem to be inconsistent with our naturalist world view. But these philosophers still face the problem that any scientific analysis of mental states and linguistic expressions will employ mathematics as part of the analysis. Since mathematical language appears to be committed to mathematical objects, the scientific analyses arouse suspicions similar to those analyses mentioned at the outset, for mathematical objects are not obviously part of the natural world. But in the analysis of mind, one has to account for the content of mental states about mathematical objects, as well as for states of affairs that don't obtain, thougts about fictions, etc., and similarly, in the analysis of the content (denotation, meaning, truth conditions) of linguistic expressions.
In this talk, I discuss a way to naturalize the puzzling, "non-natural" entities (abstract objects, properties and propositions, possibilities, and mathematical objects) by way of a new interpretation of the precise principles which assert their existence. These principles assert the existence of objects and properties in a way that corresponds to certain expressible conditions in a formal language. If we interpret these principles as ones which *systematize our linguistic practices*, then a path opens up to the naturalization of the puzzling entities. I travel down this path and explore its consequences. The most interesting consequence is that a somewhat imprecise Wittgensteinian view of meaning becomes wedded to some precise formal machinery, clarifying both in the process. Philosophers of mind need not eschew the logical techniques used by logicians and philosophers of language, since there is a way to reconcile these logical analyses with our naturalistic world view.
Title: Convergence in the Philosophy of Mathematics
Abstract: The Platonist answer to the question, "What is mathematical language about?", is that it is about abstract individuals (such as zero, the null set, omega, etc.) and abstract relations (such successor, less than, set membership, group addition, etc.). One way to give a metaphysical foundation for mathematics is to support this answer with an axiomatic theory of abstract individuals and abstract relations and an analysis of mathematical language that yields denotations for the terms of mathematical theories and truth conditions for mathematical claims. I'll review such a theory and then show that the background *formalism* for expressing the theory is subject to various interpretations. The Platonistic interpretation is just one of the many ways of interpreting the formalism and the analysis of mathematics. I'll show that one can develop fictionalist, structuralist, inferentialist, if-thenist, finitist, and logicist interpretations of the formalism. Since each interpretation offers us a clear, but different, answer to our initial question, the resulting analysis not only offers a way to make these philosophies of mathematics more precise, but also unifies them in a new and unsuspected way.
Title: Foundations for Mathematical Structuralism (coauthor: Uri Nodelman)
Abstract: We investigate the form of mathematical structuralism that acknowledges the existence of structures and their distinctive structural elements. This form of structuralism has been subject to criticisms recently, and our view is that the problems raised are resolved by proper, mathematics-free theoretical foundations. Starting with an axiomatic theory of abstract objects, we identify a mathematical structure as an abstract object encoding the truths of a mathematical theory. From such foundations, we derive consequences that address the main questions and issues that have arisen. Namely, elements of different structures are different. A structure and its elements ontologically depend on each other. There are no haecceities and each element of a structure must be discernible within the theory. These consequences are not developed piecemeal but rather follow from our definitions of basic structuralist concepts.
Title: A Defense of Logicism (coauthor: Hannes Leitgeb)
Abstract: In this talk, we extend the argument in the paper "What is Neologicism?" (B. Linsky and E. Zalta, Bulletin of Symbolic Logic, 12/1 (June 2006): 60–99). Linsky and Zalta argued that if the notion of reduction used by the original logicists is *weakened*, a new [sic] form of neologicism emerges that can be generally applied to arbitrary mathematical theories. In the present talk, however, we develop positive arguments for thinking: (1) that the notion of reduction assumed by the early logicists is the *wrong* notion of reduction given their epistemological motivations and goals; (2) that the notion of "ontological reduction" defined in "Neo-logicism? An Ontological Reduction of Mathematics to Metaphysics" (Erkenntnis, 53/1-2 (2000): 219--265) allows one to attain the epistemological goals driving logicism; (3) that when the comprehension principle for object theory is replaced by the equipotent abstraction principle, the resulting system is a logic, given that we accept that weak second-order logic is indeed (part of) logic even if full second-order logic is not; and thus (4) logicism is true: since arbitrary mathematical theories are ontologically reducible in the logic of object theory, mathematics is reducible to logic.
Title: The Fundamental Theorem of World Theory (coauthor: Christopher Menzel)
Abstract: Each conception of possible worlds is defined by the principles that govern them. The most fundamental principle of Lewis's conception of worlds, for example, says: (absolutely) every way a world might be is a way that some world is (On the Plurality of Worlds: 2, 71, 86). This "Lewis Principle" grounds any reasonable theory of possible worlds, including those based on a more abstract conception of them. We review how a representation of this principle can be derived in object theory, and then show that the axioms from which it is derived are true in tiny models. (The last fact is confirmed using tools of computational metaphysics.) So Lewis's *theoretical* principle (prior to application) doesn't require a large ontology. We then argue that the axioms used in the derivation of the Lewis Principle are logical in nature and are analytic, by analogy with other principles that we accept as logical and analytic. From this, one can argue that belief in possible worlds is more easily justified. We don't have to justify belief in worlds on a case-by-case basis; instead, we need only justify the Lewis Principle, and we can do so on the grounds that it is derivable from analytic truths.
Title: Infinity Without Mathematical Primitives or Axioms
Abstract: In this talk, I show how the existence of numbers, an infinite number, and an infinite set can be proved without any mathematical primitives or mathematical axioms. The derivation takes place in the extended theory of abstract objects. In contrast to ZF, we don’t assert an axiom of infinity, and in contrast to second-order logic extended by Hume's Principle, we don't take #F as a primitive notion or Hume’s Principle as an axiom. Nevertheless, some key parts of the strategy used in Frege's Theorem are preserved. The results are not intended as a replacement for any part of mathematics, but rather as a study into a body of logical concepts and principles from which something infinite emerges.
Title: A Computationally-Discovered Simplification of the Ontological Argument (coauthor: Paul E. Oppenheimer)
Abstract: Computational techniques show that there is a simplification of Anselm's ontological argument. We begin by reviewing our 1991 paper ("On the Logic of the Ontological Argument", reprinted in the Philosopher's Annual 1993). In that paper, we represent Anselm's argument in the Proslogion using logical axioms and theorems governing definite descriptions and 3 non-logical premises (relating "greater than", "conceivability" and "existence"). However, when we represent these theorems and premises in Prover9 the latter discovers an argument for the existence of God that shows Anselm needs only a single non-logical premise in addition to the logical machinery of definite descriptions. We reverse engineer the Prover9 argument into a more easily recognizable form, and thereby show how one non-logical premise involving the description "the conceivable thing such that nothing greater is conceivable" can imply the claim that God exists by way of a simple *diagonal* type argument. The soundness of Anselm's argument thus rests on a single claim, and the analysis of that claim concludes the talk.
Abstract: In this talk, I present new results of the computational metaphysics project, developed in conjunction with the other project members Jesse Alama and Paul Oppenheimer. The computational metaphysics project was initially described in the paper "Steps Toward Computational Metaphysics" (Branden Fitelson and Edward N. Zalta), published in J. Philosophical Logic, 36/2 (April 2007): 227--247). In this research, we extend these basic results using such automated reasoning engines as the E prover system and VAMPIRE, the PARADOX model-building program. I demonstrate some command-line tools developed by Alama for determining, from a set of premises that prove a conclusion, the smallest subset or subsets of those premises needed to prove the conclusion. I focus primarily on the implementation of object theory's analysis of Leibniz's non-modal and modal calculus of concepts. As part of the talk, I review the basic principles of axiomatic theory of abstract objects.