Stewart Shapiro

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.

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?

Michael Lynch’s Truth as One and Many is a contribution to the large body of philosophical literature on the nature of truth. Within that genre, advocates of truth-as-correspondence, advocates of truth-as-coherence, and the like, all hold that truth has a single underlying metaphysical nature, but they sharply disagree as to what this nature is. Lynch argues that many of these views make good sense of truth attributions for a limited stretch of discourse, but he adds that each of the contenders also faces a ‘scope problem’, in that none of them makes sense of all legitimate truth attributions. Some deflationists make similar observations – usually less concessive – and conclude that truth simply has no substantial, underlying nature. There is nothing for the philosopher to articulate and defend. Lynch agrees with deflationists that the realm of discourse is too diverse for any single ‘first-level’ property, such as correspondence or coherence, to fully characterize it, but against deflationism, he holds that truth does have a nature. He presents a functionalist theory. The definition is this: x is true if, and only if, x has a property that plays the truth-role. According to Lynch, the truth-role is given in terms of what he calls the ‘core truisms’ of the folk concept of truth. These are platitudes that hold of the everyday notion, and are such that anyone who denies all of them can be accused of changing the subject – of not talking about truth. A given property T plays the truth-role for a batch of propositions, just in case, for any proposition P in the batch: P is T if, and only if, where P is believed, things are as they are believed to be; other things being equal, it is a worthy goal …

1. Philosophical background: iteration, ineffability, reflection. There are at least two heuristic motivations for the axioms of standard set theory, by which we mean, as usual, first-order Zermelo–Fraenkel set theory with the axiom of choice (ZFC): the iterative conception and limitation of size (see Boolos, 1989). Each strand provides a rather hospitable environment for the hypothesis that the set-theoretic universe is ineffable, which is our target in this paper, although the motivation is different in each case.

Around the turn of the century, Poincare and Hilbert each published an account of geometry that took the discipline to be an implicit definition of its concepts. The terms ‘point’, ‘line’, and ‘plane’ can be applied to any system of objects that satisfies the axioms. Each mathematician found spirited opposition from a different logicist—Russell against Poincare' and Frege against Hilbert— who maintained the dying view that geometry essentially concerns space or spatial intuition. The debates illustrate the emerging idea of mathematics as the science of structure. The article closes with some remarks on structuralist and nonstructuralist themes in Frege's own development of arithmetic.