David Hilbert e o axioma de Arquimedes: entre a geometria e a física

Resumo

*Doutor em Lógica e Filosofia da Ciência pela Unicamp. Professor no Instituto de Filosofia da Universidade Federal de Uberlândia (UFU).

David Hilbert e o axioma de Arquimedes: entre a geometria e a física

Resumo: A relação entre geometria e física na obra de Hilbert é analisada através do caso do Axioma de Arquimedes. Começando com as questões geométricas e as formais, em particular a definição de modelos não arquimedeanos para provar a independência, passa-se logo à concepção de Hilbert da geometria como uma ciência empírica, para depois estudar a afirmação de Hilbert de que o Axioma de Arquimedes deve ser testado empiricamente. Nesse sentido, esse autor enuncia uma formulação empírica do axioma, a qual, segundo afirma, deveria ser submetida à experimentação. Tal enunciado coloca três tipos de questões. Primeiro, se é realmente um enunciado empírico ou se é um princípio metodológico que não pode ser testado. Segundo, se é uma interpretação adequada desse axioma. Por último, como poderiam ser ideali­zados testes a partir desse enunciado. Sendo as primeiras questões problemáticas, pior é o caso da terceira, pois resulta difícil conceber experimentos bem definidos nos quais a formulação empírica possa ser testada, diferente, por exemplo, do caso da medição dos ângulos de um triângulo entre três picos encomendada por Gauss. Na estudo da formulação empírica também são analisados os comentários de Leo Corry e de Michael Stöltzer sobre o assunto, resultando em questionamentos sobre sua adequação e verificabilidade. Além disso, é salientada a importância de diferenciar os conceitos de mensuração, próprio do Axioma de Arquimedes, e de continuidade no sentido definido por Dedekind, baseado fundamentalmente na crítica que Sommer faz a Hilbert.

David Hilbert e o axioma de Arquimedes: entre a geometria e a física

Abstract: The relationship between geometry and physics in the work of Hilbert is analyzed through the case of the axiom of Archimedes. Starting with geometrical and formal issues (in particular, the definition of non-Archimedean models used for proving its independence), following with Hilbert conception that the physics is an empirical science, finally it is studied the Hilbertian statement that the axiom must be empirically confirmed. In this sense, he conceives an empirical statement of the axiom which must be confirmed by experiment. This statement arises three kind of questions. First, whether it actually is an empirical statement or a methodological rule which is not able to test. Second, whether it is a suitable interpretation of the axiom. Third, how can be created tests in relation to the empirical statement. Besides the fact that the two initial questions are difficult, the case of the third is worst, because, as far as I know, nobody proposed such a test, i.e. how can be designed well defined experiments to confirm the empirical statement (different, for example, of the case of the measurement of the sum of the angles of a triangle, performed by Gauss). The criticisms of Leo Corry and Michael Stöltzer are analyzed too, in particular the questions about adequacy and verification of the empirical statement. Furthermore, it is emphasized the relevance of the distinction between the concepts of measurement, inherent to the Archimedean axiom, and the one of the continuity (in Dedekind's sense), based upon the criticism of Sommer on the Foundations of Geometry of Hilbert.

Keywords: Hilbert. Axiom of Archimedes. Geometry. Physics.

Data de registro: 11/10/2013

Data de aceite: 23/04/2014

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