INTRODUCTION

Between 1897 and 1902, there took place a brief correspondence between Frege and Hilbert, consisting of four letters from Frege, and two letters and three postcards from Hilbert. It centres on Frege's reactions to Hilbert's classic Grundlagen der Geometrie, first published in 1899, and Hilbert's restatements in his letters to Frege of the foundational positions which that work, sometimes only implicitly, embodies. Despite the obvious richness of common purpose between Frege and Hilbert, the correspondence is especially instructive because of the strong disagreements expressed. For example, the two disagreed on the form and function of definitions, the nature, purpose and formulation of axioms, the nature of (axiomatized) mathematical theories, the method of independence proofs in geometry, the role and form of consistency proofs and the nature of mathematical existence. Many of the articles of disagreement, especially those on axioms and independence proofs, also reveal or underline significant differences in their respective conceptions of logic. Frege followed the correspondence with two polemical, and wider-ranging, articles on similar or related themes, Hilbert himself having apparently declined Frege's suggestion that their exchange of views be published. These two papers help to fill out the picture on Frege’s side, first by restating Frege’s opposition, and then by presenting his insights into the formal structure of Hilbert’s position. Especially important are Frege’s attempts in his second article to render central results of Hilbert’s project as read through his own system.

INTRODUCTION

It is uncontroversial to say that the period in question saw more important changes in the philosophy of mathematics than any previous period of similar length in the history of philosophy. Above all, it is in this period that the study of the foundations of mathematics became partly a mathematical investigation itself. So rich a period is it, that this survey article is only the merest sketch; inevitably, some subjects and figures will be inadequately treated (the most notable omission being discussion of Peano and the Italian schools of geometry and logic). Of prime importance in understanding the period are the changes in mathematics itself that the nineteenth century brought, for much foundational work is a reaction to these, resulting either in an expansion of the philosophical horizon to incorporate and systematise these changes, or in articulated opposition. What, in broad outline, were the changes?

First, traditional subjects were treated in entirely new ways. This applies to arithmetic, the theory of real and complex numbers and functions, algebra, and geometry. (a) Some central concepts were characterised differently, or properly characterised for the first time, for example, from analysis, those of continuity (Weierstrass, Cantor, Dedekind) and integrability (Jordan, Lebesgue, Young), from geometry, that of congruence (Pasch, Hilbert), and geometry itself was recast as a purely synthetic theory (von Staudt, Pasch, Hilbert). (b) Theories were treated in entirely new ways, for example, as axiomatic systems (Pasch, Peano and the Italian School, Hilbert), as structures (Dedekind, Hilbert), or with entirely different primitives (Riemann, Cantor, Frege, Russell).

Introduction

The aquatic and terrestrial components of riparian systems provide ecological opportunities for many species of mammals. The importance of riparian habitat to wildlife populations has been documented in a wide range of habitats in North America: the midwestern United States (Stauffer and Best 1980), desert southwest (England et al. 1984), Rocky Mountains (Knopf 1985), Oregon (Anthony et al. 1987, Doyle 1990, McComb et al. 1993, Gomez and Anthony 1998, Kauffman et al. 2001), Washington (O'Connell et al. 1993, Kelsey and West 1998, Kauffman et al. 2001), and the Okanogan Highlands of British Columbia (Gyug 2000). These studies indicate that wildlife species richness is high in these ecosystems, and use of riparian zones by some species is disproportionately higher than in other areas. Although this is especially true in the more arid regions of North America (Johnson and Jones 1977, Brinson et al. 1981), this pattern can also be found in mesic forests of the Pacific Northwest. For example, Thomas et al. (1979) report that 285 of the 378 terrestrial wildlife species in the Blue Mountains of Oregon and Washington are found exclusively or more commonly in riparian areas, and Oakley et al. (1985) report similar patterns of 359 of the 414 wildlife species using riparian zones of western Washington and Oregon forests. Kauffman et al. (2001) estimate that 53% of the 593 wildlife species that occur in Washington and Oregon use riparian zones, whereas riparian zones and wetlands constitute only 1% to 2% of the landscape.