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.

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.

John Searle began to discuss his recently published book `The Construction of Social Reality' with Anthony Freeman, and they ended up talking about God. The book itself and part of their conversation are introduced and briefly reflected upon by Anthony Freeman. Many familiar social facts -- like money and marriage and monarchy -- are only facts by human agreement. They exist only because we believe them to exist. That is the thesis, at once startling yet obvious, that philosopher John Searle (...) explores in The Construction of Social Reality. (shrink)

Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also analyzes the concept of intuition (...) and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite. (shrink)

From Brouwer To Hilbert: The Debate on the Foundations of Mathematics in the 1920s offers the first comprehensive introduction to the most exciting period in the foundation of mathematics in the twentieth century. The 1920s witnessed the seminal foundational work of Hilbert and Bernays in proof theory, Brouwer's refinement of intuitionistic mathematics, and Weyl's predicativist approach to the foundations of analysis. This impressive collection makes available the first English translations of twenty-five central articles by these important contributors and many others. (...) The articles have been translated for the first time from Dutch, French, and German, and the volume is divided into four sections devoted to Brouwer, Weyl, Bernays and Hilbert, and the emergence of intuitionistic logic. Each section opens with an introduction which provides the necessary historical and technical context for understanding the articles. Although most contemporary work in this field takes its start from the groundbreaking contributions of these major figures, a good, scholarly introduction to the area was not available until now. Unique and accessible, From Brouwer To Hilbert will serve as an ideal text for undergraduate and graduate courses in the philosophy of mathematics, and will also be an invaluable resource for philosophers, mathematicians, and interested non-specialists. (shrink)

Mit den "Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie" von 1913, von ihm selbst nur als eine "Allgemeine Einführung in die reine Phänomenologie" angezeigt, zog Edmund Husserl die Konsequenz aus seinen Logischen Untersuchungen (PhB 601), die ihn 1900/01 berühmt gemacht hatten: Ausgehend von der dort entwickelten Phänomenologie der intentionalen Erlebnisse sieht er jetzt in der Aufdeckung der Leistungen des "reinen Bewußtseins", dem die uns bekannte natürliche Welt nur als "Bewußtseinskorrelat" gegeben ist, den eigentlichen Gegenstand philosophischer Erkenntnis und in den (...) von ihm eingeführten methodologischen Begriffen der "Reduk tion" und der "Epoché" den Weg, sich über die Beschaffenheit dieses "reinen Bewußtseins", aus dem alle Erkenntnis entspringt, Klarheit zu verschaffen. Die Ausgabe bietet Husserls Schrift text- und seitengleich nach Band III/1 der Husserliana, herausgegeben von Karl Schuhmann; beigegeben sind Husserls Nachwort zu den Ideen von 1930 aus Band V der Husserliana, herausgegeben von Marly Biemel, sowie ein umfangreiches Namen- und Sachregister von Elisabeth Ströker. (shrink)

This paper is mainly concerned with proof-theoretic analysis of some second-order systems of explicit mathematics with a non-constructive minimum operator. By introducing axioms for variable types we extend our first-order theory BON to the elementary explicit type theory EET and add several forms of induction as well as axioms for μ. The principal results then state: EET plus set induction is proof-theoretically equivalent to Peano arithmetic PA <0).

This massive two-volume reference presents a comprehensive selection of the most important works on the foundations of mathematics. While the volumes include important forerunners like Berkeley, MacLaurin, and D'Alembert, as well as such followers as Hilbert and Bourbaki, their emphasis is on the mathematical and philosophical developments of the nineteenth century. Besides reproducing reliable English translations of classics works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare, William Ewald also includes selections from Gauss, Cantor, Kronecker, and Zermelo, all translated here for (...) the first time. (shrink)

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

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)

In this sequence of philosophical essays about natural science, the author argues that fundamental explanatory laws, the deepest and most admired successes of modern physics, do not in fact describe regularities that exist in nature. Cartwright draws from many real-life examples to propound a novel distinction: that theoretical entities, and the complex and localized laws that describe them, can be interpreted realistically, but the simple unifying laws of basic theory cannot.

We show that the following results in the classical theory of unbounded linear operators on Hilbert spaces can be proved within the framework of Bishop's constructive mathematics: the Kato-Rellich theorem, the spectral theorem, Stone's theorem, and the self-adjointness of the most common quantum mechanical operators, including the Hamiltonians of electro-magnetic fields with some general forms of potentials.

Hermann Weyl, one of the twentieth century's greatest mathematicians, was unusual in possessing acute literary and philosophical sensibilities—sensibilities to which he gave full expression in his writings. In this paper I use quotations from these writings to provide a sketch of Weyl's philosophical orientation, following which I attempt to elucidate his views on the mathematical continuum, bringing out the central role he assigned to intuition.

In this collection of essays written over a period of twenty years, Solomon Feferman explains advanced results in modern logic and employs them to cast light on significant problems in the foundations of mathematics. Most troubling among these is the revolutionary way in which Georg Cantor elaborated the nature of the infinite, and in doing so helped transform the face of twentieth-century mathematics. Feferman details the development of Cantorian concepts and the foundational difficulties they engendered. He argues that the freedom (...) provided by Cantorian set theory was purchased at a heavy philosophical price, namely adherence to a form of mathematical platonism that is difficult to support. -/- Beginning with a previously unpublished lecture for a general audience, Deciding the Undecidable, Feferman examines the famous list of twenty-three mathematical problems posed by David Hilbert, concentrating on three problems that have most to do with logic. Other chapters are devoted to the work and thought of Kurt Gödel, whose stunning results in the 1930s on the incompleteness of formal systems and the consistency of Cantors continuum hypothesis have been of utmost importance to all subsequent work in logic. Though Gödel has been identified as the leading defender of set-theoretical platonism, surprisingly even he at one point regarded it as unacceptable. -/- In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part--if not all--of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, Is Cantor Necessary?, is answered with a resounding no. -/- This volume of important and influential work by one of the leading figures in logic and the foundations of mathematics is essential reading for anyone interested in these subjects. (shrink)

This fascinating exploration of the great discoveries of history's most important mathematicians seeks an answer to the eternal question: Does mathematics hold the key to understanding the mysteries of the physical world?

This book analyzes the different ways mathematics is applicable in the physical sciences, and presents a startling thesis--the success of mathematical physics ...

Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.

The present volume is intended as an all-round introduction to constructivism. Here constructivism is to be understood in the wide sense, and covers in particular Brouwer's intuitionism, Bishop's constructivism and A.A. Markov's constructive recursive mathematics. The ending "-ism" has ideological overtones: "constructive mathematics is the (only) right mathematics"; we hasten, however, to declare that we do not subscribe to this ideology, and that we do not intend to present our material on such a basis.

Stephen George Simpson. with definition 1.2.3 and the discussion following it. For example, taking 90(n) to be the formula n §E Y, we have an instance of comprehension, VYEIXVn(n€X<—>n¢Y), asserting that for any given set Y there exists a ...

This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabhäuser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions of independence of axioms and primitive notions, and versions of the system suitable for the development of 1-dimensional geometry.

This is the first elementary book to employ the concept of infinitesimals.

On the effectiveness and limits of mathematics in physics / A.O. Barut -- Why is the universe knowable? / P.C.W. Davies -- Mathematics in sociology: Cinderella's carriage or pumpkin? / Patrick Doreian -- Fundamental roles of mathematics in science / Donald Greenspan -- Inner vision, outer truth / Reuben Hersh -- Mathematics and the natural order / Wendell G. Holladay -- A few systems-colored views of the world / Yi Lin -- The reasonable effectiveness of mathematical reasoning / Saunders Mac (...) Lane -- Three aspects of the effectiveness of mathematics in science / Louis Narens and R. Duncan Luce -- Mathematics and natural philosophy / Robert L. Oldershaw -- Mathematics and the language of nature / F. David Peat -- The reason within and the reason without / John Polkinghorne -- The modelling relation and natural law / Robert Rosen -- Structure and effectiveness / La Verne Shelton -- Psychology and mathematics / James T. Towns end and Helena Kadlec -- Ariadne's thread: the role of mathematics in physics / Hans C. von Baeyer -- The disproportionate response / Bruce J. West -- The unreasonable effectiveness of mathematics in the natural sciences / Eugene P. Wigner (reprint) -- The effectiveness of mathematics in fundamental physics / A. Zee. (shrink)