Results 1  10
of
14
Does category theory provide a framework for mathematical structuralism?
 PHILOSOPHIA MATHEMATICA
, 2003
"... Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à 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 ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à 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 “manytopoi” view and modalstructuralism. 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
Predicative Foundations of Arithmetic
 Journal of Philosophical Logic
, 1995
"... Predicative mathematics in the sense originating with Poincaré andWeylbegins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or settheoretic standpoint, this appears problematic, for, as the story is usu ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
Predicative mathematics in the sense originating with Poincaré andWeylbegins by taking the natural number system for granted, proceeding immediately to real analysis and related fields. On the other hand, from a logicist or settheoretic standpoint, this appears problematic, for, as the story is usually told, impredicative
Scientific Representation and the Semantic View of Theories”, Theoria 55: 49–65
, 2006
"... ABSTRACT: It is now part and parcel of the official philosophical wisdom that models are essential to the acquisition and organisation of scientific knowledge. It is also generally accepted that most models represent their target systems in one way or another. But what does it mean for a model to re ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
ABSTRACT: It is now part and parcel of the official philosophical wisdom that models are essential to the acquisition and organisation of scientific knowledge. It is also generally accepted that most models represent their target systems in one way or another. But what does it mean for a model to represent its target system? I begin by introducing three conundrums that a theory of scientific representation has to come to terms with and then address the question of whether the semantic view of theories, which is the currently most widely accepted account of theories and models, provides us with adequate answers to these questions. After having argued in some detail that it does not, I conclude by pointing out in what direction a tenable account of scientific representation might be sought.
8 Circularity and Paradox
, 2004
"... Both the paradoxes Ramsey called semantic and the ones he called settheoretic look to be paradoxes of circularity. What does it mean to say this? I suppose it means that they look to turn essentially on circular notions of the relevant disciplines: semantics and set theory. But what does it mean to ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Both the paradoxes Ramsey called semantic and the ones he called settheoretic look to be paradoxes of circularity. What does it mean to say this? I suppose it means that they look to turn essentially on circular notions of the relevant disciplines: semantics and set theory. But what does it mean to call a notion circular? I suppose that a circular notion (of discipline D) is one of the form selfR, for R a key relation of that discipline. Reference and predication are key semantic relations, so selfreference and selfpredication are circular notions of semantics. Membership is a key settheoretic relation, so selfmembership is a circular notion of set theory. The set and semantic paradoxes look to be paradoxes of circularity because they look to turn essentially on notions like selfmembership and selfreference. This approach to circularity might seem insufficiently discriminating. Do we really want to count selfdeception and selfincrimination in with selfreference and selfmembership? Well, why not? Remember, the target here is not circular notions as such but circularitybased paradox. We get a circularitybased paradox when a circular notion generates absurdities, with the circularity of the notion playing an essential role. I don’t know whether selfdeception and selfincrimination generate absurdities in this way. But if they do, then I for one am happy to speak of circularitybased paradoxes of psychoanalysis or legal theory.
Russell’s Absolutism vs.(?)
"... Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And ..."
Abstract
 Add to MetaCart
Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind’s “structuralist” views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern structuralism and their challenges are reviewed, taking account of Russell’s insights. Problems of absolutism plague some versions, and, interestingly, Russell’s critique of Dedekind can be extended to one of them, ante rem structuralism. This leaves modalstructuralism and a category theoretic approach as remaining nonabsolutist
Statement
"... legal responsibility for the information which this document contains or the use to which this information is subsequently put. Although every step is taken to ensure that the information is as accurate as possible, it is understood that this material is supplied on the basis that there is no legal ..."
Abstract
 Add to MetaCart
legal responsibility for the information which this document contains or the use to which this information is subsequently put. Although every step is taken to ensure that the information is as accurate as possible, it is understood that this material is supplied on the basis that there is no legal responsibility for these materials or resulting from the use to which these can or may be put. Note: the telephone and fax numbers given in this guide for addresses outside the United Kingdom are those to be used if you are in that country. If you are telephoning or faxing from another country, we suggest you contact your local telecommunications provider for details of the country code and area code that you should use. Main contents
A Nominalist’s Dilemma and its Solution
 PHILOSOPHIA MATHEMATICA
, 2005
"... Current versions of nominalism in the philosophy of mathematics have the benefit of avoiding commitment to the existence of mathematical objects. But this comes with the cost of not taking mathematical theories literally. Jody Azzouni’s Deflating Existential Consequence has recently challenged this ..."
Abstract
 Add to MetaCart
Current versions of nominalism in the philosophy of mathematics have the benefit of avoiding commitment to the existence of mathematical objects. But this comes with the cost of not taking mathematical theories literally. Jody Azzouni’s Deflating Existential Consequence has recently challenged this conclusion by formulating a nominalist view that lacks this cost. In this paper, we argue that, as it stands, Azzouni’s proposal does not yet succeed. It faces a dilemma to the effect that either the view is not nominalist or it fails to take mathematics literally. After presenting the dilemma, we suggest a possible solution for the nominalist.
AXIOMATIZING MATHEMATICAL CONCEPTUALISM IN THIRD ORDER ARITHMETIC
"... Abstract. We review the philosophical framework of mathematical conceptualism as an alternative to settheoretic foundations and show how mainstream mathematics can be developed on this basis. The paper includes an explicit axiomatization of the basic principles of conceptualism in a formal system C ..."
Abstract
 Add to MetaCart
Abstract. We review the philosophical framework of mathematical conceptualism as an alternative to settheoretic foundations and show how mainstream mathematics can be developed on this basis. The paper includes an explicit axiomatization of the basic principles of conceptualism in a formal system CM set in the language of third order arithmetic. This paper is part of a project whose goal is to make a case that mathematics should be disassociated from set theory. The reasons for wanting to do this, which I discuss in greater detail elsewhere ([22]; see also [19] and [23]), involve both the philosophical unsoundness of set theory and its practical irrelevance to mainstream mathematics. Set theory is based on the reification of a collection as a separate object, an elementary philosophical error. Not only is this error obvious, it also has the spectacular consequence of immediately giving rise to the classical set theoretic paradoxes. Of course, these paradoxes are not derivable in the standard axiomatizations of set theory, but that is only because these systems were specifically designed to avoid them. In these systems the paradoxes are blocked by means of ad hoc restrictions on the set concept that have no obvious intuitive justification, which has led to the development of a large literature of attempted rationalizations
TWODIMENSIONAL BELIEF CHANGE An Advertisement
"... In this paper I compare two different the models of twodimensional belief change, namely ‘revision by comparison ’ (Fermé and Rott, Artificial Intelligence 157, 2004) and ‘bounded revision ’ (Rott, in Hommage à Wlodek, Uppsala 2007). These revision operations are twodimensional in the sense that t ..."
Abstract
 Add to MetaCart
In this paper I compare two different the models of twodimensional belief change, namely ‘revision by comparison ’ (Fermé and Rott, Artificial Intelligence 157, 2004) and ‘bounded revision ’ (Rott, in Hommage à Wlodek, Uppsala 2007). These revision operations are twodimensional in the sense that they take as arguments pairs consisting of an input sentence and a reference sentence. Twodimensional revision operations add a lot to the expressive power of traditional qualitative approaches to belief revision and refrain from assuming numbers as measures of degrees of belief. 1.
Philosophia Mathematica (III) 13 (2005), 294–307. doi:10.1093/philmat/nki033 A Nominalist’s Dilemma and its Solution †
"... the benefit of avoiding commitment to the existence of mathematical objects. But this comes with the cost of not taking mathematical theories literally. Jody Azzouni’s Deflating Existential Consequence has recently challenged this conclusion by formulating a nominalist view that lacks this cost. In ..."
Abstract
 Add to MetaCart
the benefit of avoiding commitment to the existence of mathematical objects. But this comes with the cost of not taking mathematical theories literally. Jody Azzouni’s Deflating Existential Consequence has recently challenged this conclusion by formulating a nominalist view that lacks this cost. In this paper, we argue that, as it stands, Azzouni’s proposal does not yet succeed. It faces a dilemma to the effect that either the view is not nominalist or it fails to take mathematics literally. After presenting the dilemma, we suggest a possible solution for the nominalist. 1.