Javier Legris

Este trabajo tiene por objetivo examinar la idea de deducción metamatemática en el programa de Hilbert, mostrando su dependencia de conceptos gnoseológicos, tales como el de conocimiento intuitivo. También se comparará esta concepcion de la deducción con la fundamentación intuicionista de la logica. Sostendré que esta deducción metamatemática lleva a una caracterización de la logica como una teoría de las deducciones formales en un sentido particular.This paper aims to examine the idea of metamathematical deduction in Hilbert’s program showing its dependence of epistemological notions, such as intuitive knowledge. This conception of deduction will be also compared with the intuitionistic foundation of logic. I will argue that this metamathematical deduction leads to a characterization of logic as a theory of formal deductions in a particular sense.

The aim of this paper is to examine the idea of metamathematical deduction in Hilbert’s program showing its dependence of epistemological notions, specially the notion of intuitive knowledge. It will be argued that two levels of foundations of deduction can be found in the last stages (in the 1920s) of Hilbert’s Program. The first level is related to the reduction – in a particular sense – of mathematics to formal systems, which are ‘metamathematically’ justified in terms of symbolic manipulation. The second level of foundation consists in warranting epistemologically the validity of the combinatory processes underlying the symbolic manipulation in metamathematics. In this level the justification was carried out with the aid of notions from modern epistemology, particularly the notion of intuition. Finally, some problems concerning Hilbert’s use of this notion will be shown and it will be compared with Brouwer’s.

The aim of this paper is to frame briefly Chateaubriand’s conception of logical forms in the distinction between logic and language as calculus and logic as universal language, devised by Jean van Heijenoort and later generalized by Jaakko Hintikka. The most important reasons to connect Chateaubriand’s conception with this distinction are perhaps Chateaubriand’s criticism of the linguistic approach to logical forms and the role Chateaubriand assigns to symbolism in his own account.O propósito deste artigo é localizar brevemente a concepção de forma lógica de Chateaubriand com relação à distinção entre lógica e linguagem como cálculo e lógica como linguagem universal, introduzida por Jean van Heijenoort e depois generalizada por Jaako Hintikka. Talvez a razão mais importante para conectar a concepção de Chateaubriand com esta distinção sejam suas críticas ao tratamento linguístico das formas lógicas, e o papel que Chateaubriand atribui ao simbolismo em seu próprio tratamento

The aim of this paper is to examine the idea of metamathematical deduction in Hilbert’s program showing its dependence of epistemological notions, specially the notion of intuitive knowledge. It will be argued that two levels of foundations of deduction can be found in the last stages (in the 1920s) of Hilbert’s Program. The first level is related to the reduction – in a particular sense – of mathematics to formal systems, which are ‘metamathematically’ justified in terms of symbolic manipulation. The second level of foundation consists in warranting epistemologically the validity of the combinatory processes underlying the symbolic manipulation in metamathematics. In this level the justification was carried out with the aid of notions from modern epistemology, particularly the notion of intuition. Finally, some problems concerning Hilbert’s use of this notion will be shown and it will be compared with Brouwer’s.

In the last century Jaakko Hintikka tried to determine Peirce’s locus within the framework of the “Logic as Calculus vs. Logic as Universal Language” opposition in the history of mathematical logic, placing it in the former tradition. For this purpose Hintikka reformulated the opposition devised earlier by Jean van Heijenoort in order to investigate not only the development of notations and formal languages in the origins of mathematical logic but also the very original ideas in them. The aim of this paper is to show some difficulties in placing Peirce’s diagrammatic conception of deductive logic inside this opposition. Firstly, Hintikka’s distinction presupposes a linguistic conception of logic by the founders of mathematical logic. However this was not Peirce’s own ultimate conception. Secondly, there is now enough evidence that Peirce conceived his diagrammatic system of the Existential Graphs mainly as a tool for logical analysis. This analysis is not of linguistic nature but rather of a semiotic one.

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.