This volume of Metatheoria includes translations into Spanish of the three famous papers on the schools in foundations of mathematics, logicism, intuitionism and formalism, presented at the Königsberg’s Symposium on Foundations of Mathematics in September 1930 and finally published in the journal Erkenntnis in 1931. The three papers constituted a milestone in the Philosophy of Mathematics of the last century. In this introduction to the translations, the editors of the volume outline the historical context in which the original papers were (...) conceived and they include some details concerning their original publication. They also stress the decisive role played by the papers in the philosophy of mathematics in the last century. A summary of them is provided and the underlying ideas of the Vienna Circle on the nature of mathematics and the impact of Gödel theorems in the foundations programs are discussed. Moreover, they briefly introduce the original contributions to the volume that follow the translations. (shrink)

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

Carnap’s work was instrumental to the liberalization of empiricism in the 1930s that transformed the logical positivism of the Vienna Circle to what came to be known as logical empiricism. A central feature of this liberalization was the deployment of the Principle of Tolerance, originally introduced in logic, but now invoked in an epistemological context in “Testability and Meaning”. Immediately afterwards, starting with Foundations of Logic and Mathematics, Carnap embraced semantics and turned to interpretation to guide the choice of a (...) theoretical language for science. The first thesis of this paper is that recourse to an intended interpretation led to a partial retrenchment of the conventionalism implied by the Principle of Tolerance. It required that the choice of a language be based on abstraction from a context; this procedure later became a component of the process of explication that was distinctive to Carnap’s mature views. The interpretive origin of formal systems also ensured their likely syntactic consistency, an issue on which Carnap was strongly criticized by figures such as Beth and Gödel. The second thesis of this paper is that this reliance on an intended interpretation enabled constructed formal systems to be relevant to the development of empirical science. (shrink)

In spite of the many efforts made to clarify von Neumann’s methodology of science, one crucial point seems to have been disregarded in recent literature: his closeness to Hilbert’s spirit. In this paper I shall claim that the scientific methodology adopted by von Neumann in his later foundational reflections originates in the attempt to revaluate Hilbert’s axiomatics in the light of Gödel’s incompleteness theorems. Indeed, axiomatics continues to be pursued by the Hungarian mathematician in the spirit of Hilbert’s school. I (...) shall argue this point by examining four basic ideas embraced by von Neumann in his foundational considerations: a) the conservative attitude to assume in mathematics; b) the role that mathematics and the axiomatic approach have to play in all that is science; c) the notion of success as an alternative methodological criterion to follow in scientific research; d) the empirical and, at the same time, abstract nature of mathematical thought. Once these four basic ideas have been accepted, Hilbert’s spirit in von Neumann’s methodology of science will become clear. (shrink)

Hans Hahn's long-neglected philosophy of mathematics is reconstructed here with an eye to his anticipation of the doctrine of logical pluralism. After establishing that Hahn pioneered a post-Tractarian conception of tautologies and attempted to overcome the traditional foundational dispute in mathematics, Hahn's and Carnap's work is briefly compared with Karl Menger's, and several significant agreements or differences between Hahn's and Carnap's work are specified and discussed.

What were the earliest reactions to Gödel's incompleteness theorems? After a brief summary of previous work in this area I analyse, by means of unpublished archival material, the first reactions in Vienna and Berlin to Gödel's groundbreaking results. In particular, I look at how Carnap, Hempel, von Neumann, Kaufmann, and Chwistek, among others, dealt with the new results.

In Honour of John Crossley’s 85th Birthday.

Symbolic logics tend to be too mathematical for the philosophers and too philosophical for the mathematicians; and their history is too historical for most mathematicians, philosophers and logicians. This paper reflects upon these professional demarcations as they have developed during the century.

An outline is given of the development of logicism from the publication of the first edition of Whitehead and Russell's Principia mathematica (1910-1913) through the contributions of Wittgenstein, Ramsey and Chwistek to Russell's own modifications made for the second edition of the work (1925) and the adoption of many of its logical techniques by the Vienna Circle. A tendency towards extensionalism is emphasised.