Each science has its own domain of investigation, but one and the same science can be formalized in different languages with different universes of discourse. The concept of the domain of a science and the concept of the universe of discourse of a formalization of a science are distinct, although they often coincide in extension. In order to analyse the presuppositions and implications of choices of domain and universe, this article discusses the treatment of omega arguments in three very different (...) formalizations of arithmetic. In Peano's formalization the domain is a restricted class of individuals, while the universe of discourse is the unrestricted class of all individuals. In Gödel's formalization the domain is a restricted class of individuals as in Peano's formalization, but the universe of discourse coincides with the domain. In Whitehead-Russell's formalization the domain is a class of logical notions in Tarski's sense, that are necessarily not individuals, whereas the universe of discourse is the unrestricted class of individuals as in Peano's formalization. The present approach emphasizes the viewpoint that the universe of discourse of a given discourse is important in determining which propositions are expressed by which sentences. (shrink)

This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics (...) is about things that really exist. (shrink)

Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense (...) of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of all natural (...) numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals. (shrink)

Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

The platonism/nominalism debate in the philosophy of mathematics concerns the question whether numbers and other mathematical objects exist. Platonists believe the answer to be in the positive, nominalists in the negative. According to non-factualists, the question is ‘moot’, in the sense that it lacks a correct answer. Elaborating on ideas from Stephen Yablo, this article articulates a non-factualist position in the philosophy of mathematics and shows how the case for non-factualism entails that standard arguments for rival positions fail. In particular, (...) showing how and why non-factualists reject nominalism illuminates the originality and interest of their position. (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)

The present collection brings together in a convenient form the seminal articles in the philosophy of mathematics by these and other major thinkers.