La filosofía del lenguaje: su naturaleza y su contexto

<p class='p1'>El artículo es un intento de determinar la naturaleza de la filosofía del lenguaje a través de las relaciones de esa disciplina con otras, tanto desde el punto de vista histórico como desde el sistemático. Se examina la forma en que la filosofía del lenguaje ha venido relacionándose de hecho con la lingüística, la lógica, la psicología y la propia filosofía, al tiempo que se hacen propuestas de clarificación de esas relaciones, en el sentido prescriptivo del término. De paso, se critican ciertas nociones que han venido oscureciendo el problema, como las de “lógica filosófica”, “filosofía lingüística” y otras, para terminar apoyando el papel fundamental de la filosofía de…

Pieri’s 1900 Paris Paper

This chapter contains an English translation of Mario Pieri’s [1900] 1901 paper Geometry Envisioned as a Purely Logical System.

Pieri’s Works on Foundations and Philosophy of Mathematics

This chapter discusses Pieri’s individual published works on foundations and philosophy of mathematics, as well as his unpublished but surviving classroom materials on projective and descriptive geometry.

Some basic theorems on the foundations of mathematics and their philosophical implications

Research in the foundations of mathematics during the past few decades has produced some results, which seem to me of interest, not only in themselves, but also with regard to their implications for the traditional philosophical problems about the nature of mathematics. The results themselves, I believe, are fairly widely known, but nevertheless, I think, it will be useful to present them in outline once again, especially in view of the fact that, due to the work of various mathematicians, they have taken on a much more satisfactory form, than they had had originally. The greatest improvement was made possible through the precise definition of the concept of finite procedure, which plays a …

The mathematics-physics analogy

The basic goal of this chapter is to delve deeply into Godel’s second great strategy in the philosophy of mathematics: the analogy between deductive and empirical sciences. Moreover, I shall try to explore the holistic, and even conventionalist implications of the analogy, such as it appears in some contemporary philosophers of mathematics who have defended the analogy to some extent. To do this, it has been necessary to present an overview of the most important precedents in the use of the analogy, such as Russell, Hilbert, Carnap, Tarski, Quine. After that, I shall present Godel’s views on the analogy, in both his published and unpublished writings. Surprisingly, most of these authors mai…

The character and origin of the manuscripts in the present edition

In the following I try to provide the reader with all the details needed to insert the Godel essays appearing here in the broader context of the rest of his unpublished work, as well as everything about my personal work on them. I start with a short description of what can be found in the Godel Nachlass in Princeton, USA, by referring to the catalogue prepared by John Dawson. Then I attempt to justify the particular selection I made of the manuscripts which I finally decided to study, reconstruct and publish. Such an explanation seems to be advisable given the great amount of Godel’s material unpublished but still extant. Also, I describe some of the historical details relevant to understan…

The analytic-synthetic distinction

This chapter tries to throw light on the first of Godel’s two main theses in the philosophy of mathematics, namely that mathematical propositions are analytic. To this end, an overview of similar conceptions is presented first in which the views by Frege, Russell Wittgenstein, Carnap and Quine are expounded. Then Godel’s view is analyzed, both in his publications and in the manuscripts which appear in this edition. The presentation of Carnap’s detailed attempt to define analyticity in his The Logical Syntax of Language (1934) may seem rather long in comparison with the ones devoted to the other authors, but it should be recalled that the Godel manuscripts appearing here were a direct philos…

Is mathematics syntax of language?, II

Around 1930 R. Carnap, H. Hahn and M. Schlick,1 largely under the influence of L. Wittgenstein, developed a conception of the nature of mathematics2 which can be characterized as being a combination of nominalism and conventionalism and which had been foreshadowed in Schlick’s doctrine about implicit definitions.3 Its main objective, according to Hahn and Schlick,4 was to conciliate strict empiricism5 with the a priori certainty of mathematics. According to this conception (which, in the sequel, I shall call the syntactical viewpoint) mathematics can completely be reduced to (or replaced by) syntax of language.6 I.e. the validity of mathematical propositions consists solely in their being c…

Realism, metamathematics, and the unpublished essays

This initial chapter is divided into two sections. The first is devoted to a brief exposition of the intuitive essence and the philosophical motivation of Godel’s main metamathematical results, namely his completeness theorem for elementary logic (1930) and his incompleteness theorems for arithmetic (1931). Thereafter some discussion of the different ways to confront the relationship between those results and Godel’s philosophical realism in logic and mathematics is offered. Thus, mathematical realism will be successively regarded as (i) a philosophical consequence of those results; (ii) a heuristic principle which leads to them; (iii) a philosophical hypothesis which is “verified” by them.…

Pieri’s Contributions to Foundations and Philosophy of Mathematics

The research of Mario Pieri (1860–1913) was widely known and valued in his lifetime. With the passage of time it largely faded from view, save for the recent attention of a relatively small number of scholars.

Pieri’s 1900 Point-and-Motion Memoir

This chapter contains an English translation of Mario Pieri’s 1900a memoir, On Elementary Geometry as a Hypothetical Deductive System: Monograph on Point and on Motion.1 By elementary geometry, Pieri meant Euclidean geometry as taught then in elementary courses, except for the theorems dependent on the Euclidean parallel axiom.

Pieri’s 1898 Geometry of Position Memoir

This chapter contains an English translation of Mario Pieri’s 1898c memoir, The Principles of the Geometry of Position Composed into a Deductive Logical System,1 his most important contribution to the foundations of projective geometry.

Frege, Peano and Russell on Descriptions: a Comparison

Pieri and Projective Geometry

Projective geometry can be described as the geometry of the straightedge, in comparison with Euclid’s geometry of straightedge and compass.

Two Paths to Logical Consequence: Pieri and the Peano School

This chapter1 has two main goals. First, it will explore the “negative” avenue leading from the concepts of independence and consistency to that of logical consequence.

Central Themes and Impact of Pieri’s Work

The first book of this series, The Legacy of Mario Pieri in Geometry and Arithmetic (cited here as M&S 2007), presented an overview of Pieri’s life and research, and a deeper study of the background of his work in foundations of geometry and arithmetic.

Pieri's Philosophy of Deductive Sciences

This chapter presents and discusses Mario Pieri’s main contributions to the philosophy of deductive, or formal, sciences.