Stewart Shapiro

The subject of this paper is the philosophical problem of accounting for the relationship between mathematics and non-mathematical reality. The first section, devoted to the importance of the problem, suggests that many of the reasons for engaging in philosophy at all make an account of the relationship between mathematics and reality a priority, not only in philosophy of mathematics and philosophy of science, but also in general epistemology/metaphysics. This is followed by a (rather brief) survey of the major, traditional philosophies of mathematics indicating how each is prepared to deal with the present problem. It is shown that (the standard formulations of) some views seem to deny outright that there is a relationship between mathematics and any non-mathematical reality; such philosophies are clearly unacceptable. Other views leave the relationship rather mysterious and, thus, are incomplete at best. The final, more speculative section provides the direction of a positive account. A structuralist philosophy of mathematics is outlined and it is proposed that mathematics applies to reality though the discovery of mathematical structures underlying the non-mathematical universe.

The is an old question over whether there is a substantial disagreement between advocates of different logics, as they simply attach different meanings to the crucial logical terminology. The purpose of this article is to revisit this old question in light a pluralism/relativism that regards the various logics as equally legitimate, in their own contexts. We thereby address the vexed notion of translation, as it occurs between mathematical theories. We articulate and defend a thesis that the notion of “same meaning” is itself context-sensitive, depending on the purposes of a given conversation.

SummaryAccording to Dummett's understanding of Frege, the sense of a denoting expression is a procedure for determining its denotation. The purpose of this article is to pursue this suggestion and develop a semi‐formal interpretation of Fregean sense for the special case of a first‐order language of arithmetic. In particular, we define the sense of each arithmetic expression to be a hypothetical process to determine the denoted number or truth value. The sense‐process is “hypothetical” in that the senses of some expressions cannot be completed in a finite number of steps. The motivation for this work is to explicate an intensional notion of effectiveness in terms of the senses of function names. It is proposed, in particular, that an expression is effective if and only if its sense can be completed in a finite number of steps. It is shown that a function is computable if and only if it has an effective name.RésuméSelon la lecture que Dummet donne de Frege, le sens ?on;une expression consiste en une procédure pour déterminer sa dénotation. Le but de cet article est de développer cette suggestion par une interprétation semi‐formelle du sens frégéen dans le cas particulier ?on;un langage de premier ordre en arithmétique. En particulier, nous définissons comme sens de toute expression mathématique un procédé hypothétique pour déterminer le nombre dénoté ou la valeur de vérité. Le procédé créateur de sens est hypothetique en ce sens que les sens de certaines expressions ne peuvent pas être donnés en un nombre fini de pas. Le but de cette étude est ?on;expliciter une notion intensionnelle ?on;effectuabilite en termes de sens des noms de fonction. II est en particulier propose qu'une expression soit dite effectuable si et seulement si son sens peut être épuisé en un nombre fini de pas. On montre qu'une fonction est calculable si et seulement si elle a un nom effectuable.ZusammenfassungDummets Auffassung von Frege gemäss besteht der Sinn eines denotierenden Ausdruckes in einem Verfahren, das dessen Denotation bestimmt. Es ist der Zweck der vorliegenden Arbeit, die‐sen Vorschlag weiterzufuUhren und eine halbformale Interpretation des Fregeschen Sinnes fiir den besonderen Fall der Sprache erster Stufe der Arithmetik zu entwickeln. Es wird im besonderen der Sinn eines jeden arithmetischen Ausdruckes als hypothetisches Verfahren zur Bestimmung der denotierten Zahl oder des Wahrheitswertes definiert. Die Sinngebung ist insofern hypothe‐tisch, als der Sinn gewisser Ausdrucke nicht in einer endlichen Zahl von Schritten gegeben wer‐den kann. Das Anliegen der Arbeit grundet in der Absicht, einen intensionalen Begriff der Effek‐tivitat aufgrund des Sinnes von Funktionsnamen zu erlautern. Es wird im besonderen festgelegt, dass ein Ausdruck dann und nur dann effektiv sei, wenn sein Sinn in einer endlichen Zahl von Schritten erreicht werden kann. Es wird gezeigt, dass eine Funktion genau dann berechenbar ist, wenn sie einen effektiven Namen hat

Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one of them is true of the other. I suggest that ‘i’ functions like a parameter in natural deduction systems. I gave an early version of this paper at a workshop on structuralism in mathematics and science, held in the Autumn of 2006, at Bristol University. Thanks to the organizers, particularly Hannes Leitgeb, James Ladyman, and Øystein Linnebo, to my commentator Richard Pettigrew, and to the audience there. The paper also benefited considerably from a preliminary session at the Arché Research Centre at the University of St Andrews. I am indebted to my colleagues Craige Roberts, for help with the linguistics literature, and Ben Caplan and Gabriel Uzquiano, for help with the metaphysics. Thanks also to Hannes Leitgeb and Jeffrey Ketland for reading an earlier version of the manuscript and making helpful suggestions. I also benefited from conversations with Richard Heck, John Mayberry, Kevin Scharp, and Jason Stanley. CiteULike    Connotea    Del.icio.us    What's this?

Since virtually every mathematical theory can be interpreted in Zermelo-Fraenkel set theory, it is a foundation for mathematics. There are other foundations, such as alternate set theories, higher-order logic, ramified type theory, and category theory. Whether set theory is the right foundation for mathematics depends on what a foundation is for. One purpose is to provide the ultimate metaphysical basis for mathematics. A second is to assure the basic epistemological coherence of all mathematical knowledge. A third is to serve mathematics, by lending insight into the various fields and suggesting fruitful techniques of research. A fourth purpose of a foundation is to provide an arena for exploring relations and interactions between mathematical fields. While set theory does better with regard to some of these and worse with regard to others, it has become the de facto arena for deciding questions of existence, something one might expect of a foundation. Given the different goals, there is little point to determining a single foundation for all of mathematics.

There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles. The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at stake when adjudicating issues concerning the identity of neo-logicist abstracts — so-called ‘Caesar questions’.

The idea that logic and reasoning are somehow related goes back to antiquity. It clearly underlies much of the work in logic, as witnessed by the development of computability, and formal and mechanical deductive systems, for example. On the other hand, a platitude is that logic is the study of correct reasoning; and reasoning is cognitive if anything Is. Thus, the relationship between logic, computation, and correct reasoning makes an interesting and historically central case study for mechanism. The purpose of this article is to begin the articulation of this relationship, pointing out its sources and its limitations.

At the beginning of Die Grundlagen der Arithmetik [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of self-evidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that both programs are undermined at a crucial point, namely when self-evidence is supported by holistic and even pragmatic considerations.

Sections 3.16 and 3.23 of Roger Penrose's Shadows of the mind (Oxford, Oxford University Press, 1994) contain a subtle and intriguing new argument against mechanism, the thesis that the human mind can be accurately modeled by a Turing machine. The argument, based on the incompleteness theorem, is designed to meet standard objections to the original Lucas-Penrose formulations. The new argument, however, seems to invoke an unrestricted truth predicate (and an unrestricted knowability predicate). If so, its premises are inconsistent. The usual ways of restricting the predicates either invalidate Penrose's reasoning or require presuppositions that the mechanist can reject

There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the slogan that mathematics is the science of structure?

We examine George Boolos's proposed abstraction principle for extensions based on the limitation-of-size conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neo-logicist development of real analysis. We show that it fails both of the conservativeness criteria for abstraction principles that Wright proposes. Thus, we support Boolos against Wright. We also show that, when combined with the axioms for Boolos's iterative notion of set, New V yields a system equivalent to full Zermelo-Fraenkel set theory with a principle of global choice. This advances Boolos's longstanding interest in the foundations of set theory.

According to ante rem structuralism a branch of mathematics, such as arithmetic, is about a structure, or structures, that exist independent of the mathematician, and independent of any systems that exemplify the structure. A structure is a universal of sorts: structure is to exemplified system as property is to object. So ante rem structuralist is a form of ante rem realism concerning universals. Since the appearance of my Philosophy of mathematics: Structure and ontology, a number of criticisms of the idea of ante rem structures have appeared. Some argue that it is impossible to give identity conditions for places in homogeneous ante rem structures, invoking a version of the identity of indiscernibles. Others raise issues concerning the identity and distinctness of places in different structures, such as the the natural number 2 and the real number 2. The purpose of this paper is to take the measure of these objections, and to further articulate ante rem structuralism to take them into account.

Since virtually every mathematical theory can be interpreted in set theory, the latter is a foundation for mathematics. Whether set theory, as opposed to any of its rivals, is the right foundation for mathematics depends on what a foundation is for. One purpose is philosophical, to provide the metaphysical basis for mathematics. Another is epistemic, to provide the basis of all mathematical knowledge. Another is to serve mathematics, by lending insight into the various fields. Another is to provide an arena for exploring relations and interactions between mathematical fields, their relative strengths, etc. Given the different goals, there is little point to determining a single foundation for all of mathematics.