In the Light of Logic

In the Light of Logic

In the Light of Logic

In the Light of Logic

Synopsis

Solomon Feferman is one of the leading figures in the philosophy of mathematics. This volume brings together a selection of his most important recent writings, covering the relation between logic and mathematics, proof theory, objectivity and intentionality in mathematics, and key issues in the work of G¿del, Hilbert, and Turing. A number of the papers appeared originally in obscure places and are not well-known, and others are published here for the first time. All of the material has been revised and annotated to bring it up to date.

Excerpt

Mathematics is extraordinarily distinctive in both its concepts and its methods. From the point of view of classical logic it is a rigid, precise, timeless, deductive "infallible" system, whereas modern anti-logical accounts stress its imprecision, fallibility, and dependency on time and culture. My own views fall between these two extremes. I believe that the anti-logical picture fails to explain what is truly distinctive about mathematics; as I see it, that can only be captured by eliciting its conceptual and logical structures. On the other hand, it is a fact that throughout its history, mathematics has been subject to fits of vagueness, uncertainty, puzzlement and, on occasion, sheer contradiction; but it is just these very problems that have made it necessary for there to be concerns about the foundations of mathematics. in the past, mathematicians dealt with such questions on a case-by-case basis in their respective disciplines, while in this century--as the nature of the problems began to cross many fields--they have largely become the province of mathematical logic.

This volume consists of a selection of my essays of an expository, historical, and philosophical character which in the main are devoted to the light logic throws on problems in the foundations of mathematics. I began writing them in the late 1970s; other pieces written over the same period which did not fit in directly with the plan chosen for this volume have been reserved for a future occasion. the present essays are grouped thematically rather than chronologically, and to some extent according to degree of accessibility to the reader. in particular, the first chapter was presented as a talk for a general audience. Beyond that, in order to give substance to the case which is made here for the essential role of logic in getting at the nature of mathematics, it is necessary to explain a number of technical concepts and results from metamathematics, that is, the logical study of formal axiomatic systems. While no knowledge of that subject on the part of the reader is presumed, a modicum of familiarity with logic would be helpful, especially in the later chapters of the volume. But I hope the general reader will find at least some of each chapter rewarding, and that those sufficiently engaged with the material will persist in reading all the way through, since there is an arc of thought which ties the problems brought out in the first part to results described later. As further assistance, an annotated list of references is included at the conclusion of chapter 1 which can be pursued in various directions, and again at various . . .

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