This article is a methodological discussion of formal approaches to the question of identity of proofs from a philosophical standpoint. First, an introduction to the question of identity of proofs itself is given, followed by a brief reconstruction of the so-called normalisation thesis, proposed by Dag Prawitz in 1971, in which some of its core mathematical and conceptual traits are presented. After that, a comparison between the normalisation thesis and the more well-known Church-Turing thesis on computability is carried out in three main parts: the first dedicated to highlighting some of the analogies between them; the second, their most remarkable differences; and the third, to the possible relations of dependence between the two. Based on these considerations, some concluding remarks concerning the potential Tiago de Castro Alves
O artigo oferece , em primeiro lugar, uma caracterização da questão da identidade de provas como um problema concernente à semântica de argumentos dedutivos, pertinente à lógica e à filosofia da linguagem. Em seguida, é dada uma descrição dos principais aspectos formais e conceituais daquela que é provavelmente a proposta de abordagem formal dessa questão mais bem de-senvolvida na literatura: a chamada tese da normalização, proposta em caráter conjectural em Prawitz 1971. Finalmente, alguns dos pressupostos semânticos que alicerçam a formulação dessa tese – componentes importantes da concepção representacionalista de provas que ela instancia – são expostos e discutidos quanto a suas possíveis motivações e consequências relevantes. Para além dos objetivos atinentes ao seu teor, este tem a meta pedagógica de tornar acessível e fomentar a consideração da questão da identidade de provas no seio da comunidade filosófica de língua portuguesa.
A topic of de Castro Alves (2019) stands in need of re-visitation, namely: possible ways of specifying restrictions on the notion of proof and some other related ones that are relevant in connection with the discussion on identity of proofs. This effort is dedicated to start a compensation for relevant shortcomings of some ideas proposed in that work. More concretely: by taking some very generic traits of proofs as a departure point, we proceed to the identification of possible outset conditions upon the investigation of identity of proofs (instead of proposing a taxonomy of criteria of identity of proofs, as in de Castro Alves (2019)). We will describe and briefly comment on two kinds of such conditions: one given in terms of how the identity of proofs is conditioned by the identity of what is proved, and other in terms of how equivalence relations between proof (re)presentations are conditioned by, on the one hand, how many distinct (collections/kinds of) proofs can be (re)presented by them, and, on the other, how many distinct (re)presentations a collection of proofs may have. To exemplify the meaningfulness of these considerations, they will be used here as basis for some critical remarks on the normalisation thesis on identity of proofs.
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