This monograph offers a fresh perspective on the applicability of mathematics in science. It explores what mathematics must be so that its applications to the empirical world do not constitute a mystery. In the process, readers are presented with a new version of mathematical structuralism. The author details a philosophy of mathematics in which the problem of its applicability, particularly in physics, in all its forms can be explained and justified. Chapters cover: mathematics as a formal science, mathematical ontology: what (...) does it mean to exist, mathematical structures: what are they and how do we know them, how different layers of mathematical structuring relate to each other and to perceptual structures, and how to use mathematics to find out how the world is. The book simultaneously develops along two lines, both inspired and enlightened by Edmund Husserl’s phenomenological philosophy. One line leads to the establishment of a particular version of mathematical structuralism, free of “naturalist” and empiricist bias. The other leads to a logical-epistemological explanation and justification of the applicability of mathematics carried out within a unique structuralist perspective. This second line points to the “unreasonable” effectiveness of mathematics in physics as a means of representation, a tool, and a source of not always logically justified but useful and effective heuristic strategies. (shrink)

The paper examines Husserl’s phenomenology and Hilbert’s view of the foundations of mathematics against the backdrop of their lifelong friendship. After a brief account of the complementary nature of their early approaches, the paper focuses on Husserl’s Formale und transzendentale Logik viewed as a response to Hilbert’s “new foundations” developed in the 1920s. While both Husserl and Hilbert share a “mathematics first,” nonrevisionist approach toward mathematics, they disagree about the way in which the access to it should be construed: Hilbert (...) wanted to reach it and show it consistent by his formalism on the basis of sensuous signs, Husserl held that there should be a reduction to elementary judgements about individuals. Husserl’s reduction does not establish the consistency of mathematics but he claims it is important for the considerations of truth. (shrink)

This is a book that no one but Weyl could have written--and, indeed, no one has written anything quite like it since.

Lecture notes from Husserl's logic lectures published during the last 20 years offer a much better insight into his doctrine of the forms of meaning than does the fourth Logical Investigation or any other work published during Husserl's lifetime. This paper provides a detailed reconstruction, based on all the sources now available, of Husserl's system of logical grammar. After having explained the notion of meaning that Husserl assumes in his later logic lectures as well as the notion of form of (...) meaning as it features in ‘doctrine of the forms of meaning’, I present a system of rules that describes all the various forms of meaning that Husserl singles out in his lectures. (shrink)

The paper discusses Husserl’s notion of definiteness as presented in his Göttingen Mathematical Society Double Lecture of 1901 as a defense of two, in many cases incompatible, ideals, namely full characterizability of the domain, i.e., categoricity, and its syntactic completeness. These two ideals are manifest already in Husserl’s discussion of pure logic in the Prolegomena: The full characterizability is related to Husserl’s attempt to capture the interconnection of things, whereas syntactic completeness relates to the interconnection of truths. In the Prolegomena (...) Husserl argues that an ideally complete theory gives an independent norm for objectivity for logic and experiential sciences, hence the notion is central to his argument against psychologism. In the Double Lecture the former is captured by non-extendibility, that is, categoricity of the domain, from which, so Husserl assumes, syntactic completeness is thought to follow. In the so-called ‘mathematical manifolds’ the expressions of the theory are equations that are reducible to equations between elements of the theory. With such an equational reduction structure individual elements of the domain are given criteria of identity and hence they are fully determined. (shrink)

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)

Erstmals lemte ich die Formale und transzendentale Logik von Edmund HusserI - damals noch in der hervorragenden englischen Ubersetzung von D. Cairns: Formal and transcendental Logic - im Herbstsemester 1977 kennen, als ich als cando phil. an der School of Philosophy der Catholic University of America in Washington, D. C., an einem Hauptseminar daruber bei Herm Prof. Dr. Robert Sokolowski teilnahm. Das an diesem Text, was mich sofort interessierte und seitdem interessiert, ist die Art und Weise, wie HusserI den "natiirIichen (...) Weg von formaler zu transzendentaler Logik zeichnet", indem er die Grundlagen der objektiven, formalen Logik des Urteils auf subjektive, transzendentale Untersuchungen tiber das Ur teilen zu sttitzen versucht. Ich wuBte zwar, daB man schon immer - ins besondere von Descartes bis Wittgenstein - davon geredet hatte, wie wichtig Evidenzfragen fUr die Logik seien, aber es schien mir, daB HusserI der erste war, der in concreto wirklich zeigen konnte, daB und inwiefem eine Theorie des Urteils einer Theorie des Urteilens, also eine Theorie der Logik einer Theorie der Erfahrung, und damit einer der Evidenz bedarf. 1m Hinblick darauf stellt das 1. Kapitel - d. h. die §§ 12-22: "Die formale Logik als apophantische Analytik" - des 1. Abschnittes der Logik den grundlegenden Text tiberhaupt dar, obwohl es nattirlich erst in enger Verbindung mit dem 4. Kapitel-d. h. mit den §§ 82-91: "RtickfUhrung der Evidenzkritik der logischen Prinzipien auf die EvideIlzkritik der Erfahrung" - des II. Abschnittes verstanden werden kann. (shrink)

In his last work, "Crisis of the European Sciences and Transcendental Phenomenology", Edmund Husserl formulated a radical new approach to phenomenological philosophy. Unlike his previous works, in the "Crisis" Husserl embedded this formulation in an ambitious reflection on the essence and value of the idea of rational thought and culture, a reflection that he considered to be an urgent necessity in light of the political, social, and intellectual crisis of the interwar period. In this book, James Dodd pursues an interpretation (...) of Husserl's text that emphasizes the importance of the problem of the origin of philosophy, as well as advances the thesis that, for Husserl, the "crisis of reason" is not a contingent historical event, but a permanent feature of a life in reason generally. (shrink)

Both critique and an appropriation of a large and diverse body of work, Home and Beyond is a major contribution to contemporary Husserl scholarship.

How can we understand the world as a whole instead of separate natural and human realms? Joseph T. Rouse proposes an approach to this classic problem based on radical new conceptions of both philosophical naturalism and scientific practice.

Machine generated contents note: Preface; Introduction: Husserl's life and writings; 1. Husserl's Crisis: an unfinished masterpiece; 2. Galileo's revolution and the origins of modern science; 3. The Crisis in psychology; 4. Rethinking tradition: Husserl on history; 5. Husserl's problematical concept of the life-world; 6. Phenomenology as transcendental philosophy; 7. The ongoing influence of Husserl's Crisis.

This book offers a philosophy for error-prone humans trying to understand messy systems in the real world.