REVISITING THE QUESTION ABOUT PROOF: PHILOSOPHICAL THEORY, HISTORY, AND MATHEMATICAL PRACTICE

Autores

  • Norma B. Goethe School of PhilosophyNational University of Cordoba

Palavras-chave:

Formal forms of reasoning. Syntactic ideal of rigor. Pragmatic dimensions of proofs. Mathematical practice. Epistemology of logic and mathematics.

Resumo

This paper revisits some of Chateaubriand’s critical considerations with regard to representing our reasoning practices in logic and mathematics by means of “idealized syntax”. I focus on the persistently critical side of these considerations which aim to prepare the ground for “an interesting epistemology of logic and mathematics” that ought to make room for understanding the pragmatic dimensions of proofs as explanatory rational displays. First, I discuss the 20th century “syntactic conception” of the logical and the underlying set of values it upholds. Secondly, I revisit the syntactic constraints on systematizing our formal forms of reasoning and ask about the relationship between “idealized” proofs construed as “syntactic objects” and the variety of formal forms of reasoning with its uses of the logical by the research mathematician. Finally, I consider the reasons why Chateaubriand thinks the syntactic requirements of “logical rigor” cannot be fulfilled, and why they ought not to be on the agenda. I conclude my paper by pointing to a deeper assumption which needs to be critically revisited as it stands in the way to what the author envisages as an “interesting epistemology of logic and mathematics”.

 

Resumo:

O presente artigo reconsidera algumas das considerações críticas de Chateaubriand com relação a representar nossa prática de raciocínios em lógica e matemática por meio de uma “sintaxe idealizada”. Concentro-me no aspecto invariavelmente crítico dessas considerações, que têm por objetivo preparar o terreno para “uma epistemologia interessante da lógica e da matemática”, a qual deve abrir caminho para a compreensão da dimensão pragmática de provas como exibição racional explicativa. Em primeiro lugar, discuto a “concepção sintática” da noção de lógica do século XX e o conjunto de valores que ela sustenta. Em segundo lugar, eu reconsidero as restrições sintáticas impostas à sistematização de nossos raciocínios formais e pergunto sobre a relação entre provas “idealizadas” construídas como “objetos sintáticos” e a variedade de modos formais de raciocínios com os seus usos do lógico pelos pesquisadores em matemática. Por fim, eu considero as razões pelas quais Chateaubriand pensa que os requisitos sintáticos do “rigor lógico” não podem ser satisfeitos, e por que eles não deveriam ser parte da agenda. Concluo meu artigo apontando uma assunção mais profunda que precisa ser reconsiderada criticamente uma vez que ela representa um obstáculo àquilo que o autor vislumbra como uma “epistemologia interessante da lógica e da matemática

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Biografia do Autor

Norma B. Goethe, School of PhilosophyNational University of Cordoba

Mathematical understanding begins with seeing. The modern view that understanding and the advancement of learning require (visual) signs or forms of expression can be traced to Leibniz. Leibniz’s insight is that language is a human creation that does not merely record our thought but is instead an embodiment of understanding, an insight that can be found also in the German mathematician-philosopher G. Frege, inventor of a two-dimensional form of writing for »the expression of pure thought« (1879).

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Publicado

2015-12-11

Como Citar

GOETHE, N. B. REVISITING THE QUESTION ABOUT PROOF: PHILOSOPHICAL THEORY, HISTORY, AND MATHEMATICAL PRACTICE. Manuscrito: Revista Internacional de Filosofia, Campinas, SP, v. 31, n. 1, p. 361–386, 2015. Disponível em: https://periodicos.sbu.unicamp.br/ojs/index.php/manuscrito/article/view/8642236. Acesso em: 8 dez. 2022.

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