The aim of this paper is to introduce Wittgenstein’s concept of the form of a language into geometry and to show how it can be used to achieve a better understanding of the development of geometry, from Desargues, Lobachevsky and Beltrami to Cayley, Klein and Poincaré. Thus this essay can be seen as an attempt to rehabilitate the Picture Theory of Meaning, from the Tractatus. Its basic idea is to use Picture Theory to understand the pictures of geometry. I will (...) try to show, that the historical evolution of geometry can be interpreted as the development of the form of its language. This confrontation of the Picture Theory with history of geometry sheds new light also on the ideas of Wittgenstein. (shrink)

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This (...) historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic. (shrink)

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This (...) historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic. (shrink)

This dissertation explores the reasons why structured graphics have been largely ignored in the representation and reasoning components of contemporary theories of axiomatic systems. In particular, it demonstrates that for the case of modern logic and geometry, there are systematic forces in the intellectual history of these disciplines which have driven the adoption of sentential representational styles over diagrammatic ones. These forces include: the changing views of the role of intuition in the procedures and formalisms of formal proof; the historical (...) contrast between the universalist subject matter of logic and the specific subject matter of geometry; and the ways in which both logic and geometry were affected by the changes that swept through mathematics during the nineteenth century. This dissertation traces the effects of each of these influences in the evolution of logic and geometry from antiquity to the early twentieth-century work of David Hilbert. For each discipline, it examines the historical factors impacting the creation and development of important diagrammatic systems, the reasons for the adoption or abandonment of these systems, and the philosophical context and debates surrounding the use of these systems. It concludes by arguing that the history of diagrammatic representations in logic and geometry cannot be adequately understood without essential reference to their philosophical background. (shrink)