There are numerous contexts in which philosophers and others use model-theoretic methods in assessing the validity of ordinary arguments; consider, for example, the use of models built upon 'possible worlds' in examinations of modal arguments. But the relevant uses of model-theoretic techniques may seem to assume controversial semantic or metaphysical accounts of ordinary concepts. So, numerous philosophers have suggested that standard uses of model-theoretic methods in assessing the validity of modal arguments commit one to accepting that modal claims are to (...) be analysed in terms of possible worlds. The paper provides a very general account of how uses of model-theoretic methods in demonstrating facts about validity and invalidity may in fact often be uncoupled from controversial semantic or metaphysical theses, applying the resulting techniques to some modal cases in particular. (shrink)

This essay is a study of ontological commitment, focused on the special case of arithmetical discourse. It tries to get clear about what would be involved in a defense of the claim that arithmetical assertions are ontologically innocent and about why ontological innocence matters. The essay proceeds by questioning traditional assumptions about the connection between the objects that are used to specify the truth-conditions of a sentence, on the one hand, and the objects whose existence is required in order for (...) the truth-conditions thereby specified to be satisfied, on the other. This allows one to set forth an assignment of truth-conditions to arithmetical sentences whereby nothing is required of the world in order for the truth-conditions of a truth of pure arithmetic to be satisfied. The essay then argues that such an assignment can be used to account for the a priori knowability of certain arithmetical truths. (shrink)

The object of this paper is to study the analogy, drawn both positively and negatively, between mathematics and fiction. The analogy is more subtle and interesting than fictionalism, which was discussed in part I. Because analogy is not common coin among philosophers, this particular analogy has been discussed or mentioned for the most part just in terms of specific similarities that writers have noticed and thought worth mentioning without much attention's being paid to the larger picture. I intend with this (...) analogy (looking at others' comparisons) to shed a little light on what is going on in mathematics, how one can understand it a bit other than experientially. This intention is philosophical and the way that I am attempting to accomplish it is also philosophical. I shall conclude my attempt to explain how it is possible and even natural for mathematics and fiction to have the analogy they have, taking it for granted, as argued in part I, that they are not to be identified. To this end I shall discuss philosophers' comparisons, mainly those of Hodes, Resnik, Tharp, and Wagner, who are the writers that seem to me to have written most thoughtfully and sufficiently extensively about fiction in making the comparison and of Körner, whose comparison is different. Whether either of these comparisons or the more general analogy is of permanent philosophical interest will have to be decided by philosophers now that they have had a fuller examination. I shall mention some other writers' reference to fiction, but not all; indeed, I am sure that I have not even found all the comparisons that there are. (shrink)

Paul Benacerraf's argument from multiple reductions consists of a general argument against realism about the natural numbers (the view that numbers are objects), and a limited argument against reductionism about them (the view that numbers are identical with prima facie distinct entities). There is a widely recognized and severe difficulty with the former argument, but no comparably recognized such difficulty with the latter. Even so, reductionism in mathematics continues to thrive. In this paper I develop a difficulty for Benacerraf's argument (...) against reductionism that is of comparable severity to the now widely recognized difficulty with his general argument against realism. Thanks to Kit Fine, Hartry Field, Jeff Sebo, Ted Sider, Stephen Schiffer, and anonymous referees at Philosophia Mathematica for helpful comments on earlier versions of this paper. Thanks to Aron Edidin for many helpful discussions of the problems that inspired it. CiteULike Connotea Del.icio.us What's this? (shrink)

ABSTRACT Historical structuralist views have been ontological. They either deny that there are any mathematical objects or they maintain that mathematical objects are structures or positions in them. Non-ontological structuralism offers no account of the nature of mathematical objects. My own structuralism has evolved from an early sui generis version to a non-ontological version that embraces Quine’s doctrine of ontological relativity. In this paper I further develop and explain this view.

This article discusses the argument we cannot have knowledge of abstract entities because they are not part of the causal order. The claim of this article is that the argument fails because of equivocation. Assume that the “causal order” is concerned with contingent facts involving time and space. Even if the existence of abstract entities is not contingent and does not involve time or space it does not follow that no truths about abstract entities are contingent or involve time or (...) space. I argue that it is the latter which is required to obtain the desired conclusion. (shrink)