Online research in philosophy

There are two central questions concerning probability. First, what are its formal features? That is a mathematical question, to which there is a standard, widely (though not universally) agreed upon answer. This answer is reviewed in the next section. Second, what sorts of things are probabilities---what, that is, is the subject matter of probability theory? This is a philosophical question, and while the mathematical theory of probability certainly bears on it, the answer must come from elsewhere. To see why, observe that there are many things in the world that have the mathematical structure of probabilities---the set of measurable regions on the surface of a table, for example---but that one would never mistake for being probabilities. So probability is distinguished by more than just its formal characteristics. The bulk of this essay will be taken up with the central question of what this “more” might be.

The paper examines the Cartesian and the Strawsonian answers to the question of why self-applied and other-applied mental predicates mean the same. While these answers relate to different, complementary aspects of this question, they seem and are usually considered as incompatible. Indeed, their apparent incompatibility constitutes a major objection to the Cartesian answer. A primary aim of the paper is to show that the Strawsonian answer does not pose a real problem to the Cartesian answer. Unlike other attempts to show this, the paper does not seek to undermine the Strawsonian answer. Indeed, its second aim is to defend this answer against these other attempts. The paper’s strategy in defending the Cartesian answer is to show that the framework underlying this answer can — indeed, for internal reasons, must — accommodate the Strawsonian answer. By showing this, the paper also shows that a Cartesian framework can provide a comprehensive answer to the aforementioned question, which is its third aim.

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be structuralists all the way down.

The debate on structuralism in the philosophy of mathematics has brought into focus a question about the status of meta-mathematics. It has been raised by Shapiro ( 2005 ), where he compares the ongoing discussion on structuralism in category theory to the Frege-Hilbert controversy on axiomatic systems. Shapiro outlines an answer according to which meta-mathematics is understood in structural terms and one according to which it is not. He finds both options viable and does not seem to prefer one over the other. The present paper reconsiders the nature of the formulae and symbols meta-mathematics is about and finds that, contrary to Charles Parsons’ influential view, meta-mathematical objects are not “quasi-concrete”. It is argued that, consequently, structuralists should extend their account of mathematics to meta-mathematics.

In this paper I argue that Category theory provides an alternative to Hilbert’s Formal Axiomatic method and doesn't support Mathematical Structuralism.

Two topics figure prominently in recent discussions of mathematical structuralism: challenges to the purported metaphysical insight provided by sui generis structuralism and the significance of category theory for understanding and articulating mathematical structuralism. This article presents an overview of central themes related to these topics.

In this paper I argue that category theory ought to be seen as providing the language for mathematical discourse. Against foundational approaches, I argue that there is no need to reduce either the content or structure of mathematical concepts and theories to the constituents of either the universe of sets or the category of categories. I assign category theory the role of organizing what we say about the content and structure of both mathematical concepts and theories. Insofar, then, as the structuralist sees mathematics as talking about structures and their morphology, I contend that category theory furnishes a framework for mathematical structuralism.

This paper considers the nature and role of axioms from the point of view of the current debates about the status of category theory and, in particular, in relation to the “algebraic” approach to mathematical structuralism. My aim is to show that category theory has as much to say about an algebraic consideration of meta-mathematical analyses of logical structure as it does about mathematical analyses of mathematical structure, without either requiring an assertory mathematical or meta-mathematical background theory as a “foundation”, or turning meta-mathematical analyses of logical concepts into “philosophical” ones. Thus, we can use category theory to frame an interpretation of mathematics according to which we can be algebraic structuralists all the way down.

In a recent paper [Hellman, 2003], we examined to what extent category theory (“CT”) provides an autonomous framework for mathematical structuralism. The upshot of that investigation was that, as it stands, while CT provides many valuable insights into mathematical structure---specific structures and structure in general---, it does not sufficiently address certain key questions of logic and ontology that, in our view, any structuralist framework needs to address. On the positive side, however, a theory of large domains was sketched as a way of supplying answers to those key questions, answers intended to be friendly to CT both in demonstrating its autonomy vis-à-vis set theory and in preserving its “arrows only” methods of describing and interrelating structures and the insights that those methods provide. The “large domains”, hypothesized as logicomathematical possibilities, are intended as suitably rich background universes of discourse relative to which both category-and-topos theory and set theory can be developed side by side, without either emerging as “prior to” the other. Although those domains, as described, resemble natural models of set theory (on an iterative conception) or toposes suitably enriched with an equivalent of the Replacement Axiom, they are defined without set-membership as a primitive, and also without ‘function’ or ‘category’ or ‘functor’ as primitives; all that is required is a combination of ‘part/whole’ and plural quantification (in effect, the resources of monadic second-order logic). This background..

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis-a-vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell's many-topoi view and modal-structuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out relative to such domains; puzzles about ‘large categories’ and ‘proper classes’ are handled in a uniform way, by relativization, sustaining insights of Zermelo.