Analysis has always been at the heart of philosophical method, but it has been understood and practised in many different ways. Perhaps, in its broadest sense, it might be defined as a process of isolating or working back to what is more fundamental by means of which something, initially taken as given, can be explained or reconstructed. The explanation or reconstruction is often then exhibited in a corresponding process of synthesis. This allows great variation in specific method, however. The aim (...) may be to get back to basics, but there may be all sorts of ways of doing this, each of which might be called ‘analysis’. The dominance of ‘analytic’ philosophy in the Englishspeaking world, and increasingly now in the rest of the world, might suggest that a consensus has formed concerning the role and importance of analysis. This assumes, though, that there is agreement on what ‘analysis’ means, and this is far from clear. On the other hand, Wittgenstein's later critique of analysis in the early (logical atomist) period of analytic philosophy, and Quine's attack on the analytic-synthetic distinction, for example, have led some to claim that we are now in a ‘post-analytic’ age. Such criticisms, however, are only directed at particular conceptions of analysis. If we look at the history of philosophy, and even if we just look at the history of analytic philosophy, we find a rich and extensive repertoire of conceptions of analysis which philosophers have continually drawn upon and reconfigured in different ways. Analytic philosophy is alive and well precisely because of the range of conceptions of analysis that it involves. It may have fragmented into various interlocking subtraditions, but those subtraditions are held together by both their shared history and their methodological interconnections. It is the aim of this article to indicate something of.. (shrink)

Wilfred Sieg. Relative Consistency and Accesible Domains.

Hilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has (...) to be seen against the background of the stunning presentation of mathematical logic in the lectures given during the winter term 1917/18. In this paper, I sketch the connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century, describe the work that laid the basis of modern mathematical logic, and analyze the first steps in the new subject of proof theory. A revision of the standard view of Hilbert's and Bernays's contributions to the foundational discussion in our century has long been overdue. It is almost scandalous that their carefully worked out notes have not been used yet to understand more accurately the evolution of modern logic in general and of Hilbert's Program in particular. One conclusion will be obvious: the dogmatic formalist Hilbert is a figment of historical (de)construction! Indeed, the study and analysis of these lectures reveal a depth of mathematical-logical achievement and of philosophical reflection that is remarkable. In the course of my presentation many questions are raised and many more can be explored; thus, I hope this paper will stimulate interest for new historical and systematic work. (shrink)

After giving a brief overview of the renewal of interest in logic and the foundations of mathematics in Göttingen in the period 1914-1921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann's doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to (...) the antinomies as found in Principia Mathematica. In the process of explaining the theory of Principia, Behmann also presented an original approach to the foundations of mathematics which saw in sense perception of concrete individuals the Archimedean point for a secure foundation of mathematical knowledge. The last part of the paper points out an important numbers of connections between Behmann's work and Hilbert's foundational thought. (shrink)

This influential 1888 publication explained the real numbers, and their construction and properties, from first principles.

The preface by Parsons begins: “This book contains the most substantial philosophical papers I wrote for publication up to 1977, with one new essay added. … The collection is unified by a common point of view underlying the essays and by certain problems that are approached from different angles in different essays. Most are directly concerned with the philosophy of mathematics, and even in those that are not … the connection between the issues discussed and mathematics is never far from (...) the surface … in the Introduction I articulate some elements of my general point of view and point out some connections between the essays.“ The nine previously published papers are unrevised except in some cases for added postscripts and footnotes. Of the essays written for publication before 1977, essay 7, “Quine on the Philosophy of Mathematics,” has not been, in fact, previously published. The eleventh, “Sets and modality,“ is the new essay. A general point of view does indeed show itself in these essays, but it is rigorously subordinated to careful and balanced analyses of the problems discussed. That is why, without sharing all elements of the point of view, I have profited so much from these essays and, seeing them collected together, am so impressed by them. (shrink)