In the framework of Stewart Shapiro, computations are performed directly on strings of symbols (numerals) whose abstract numerical interpretation is determined by a notation. Shapiro showed that a total unary function (unary relation) on natural numbers is computable in every injective notation if and only if it is almost constant or almost identity function (finite or co-finite set). We obtain a syntactic generalization of this theorem, in terms of quantifier-free definability, for functions and relations relatively intrinsically computable on certain types (...) of equivalence structures. We also characterize the class of relations and partial functions of arbitrary finite arities which are computable in every notation (be it injective or not). We consider the same question for notations in which certain equivalence relations are assumed to be computable. Finally, we discuss connections with a theorem by Ash, Knight, Manasse and Slaman which allow us to deduce some (but not all) of our results, based on quantifier elimination. (shrink)

Wroclaw’s Neo-Kantian Manifesto of Jacob FreudenthalThe initial part of the article focuses on the presentation of Jacob Freudenthal, his scientific activity and scientific achievements. The central point of the article is the analysis of the solemn speech that Jacob Freudenthal gave in front of the academic community of Wrocław on the hundredth anniversary of Immanuel Kant’s death. This speech — but also his didactic activity popularizing Kantian thought — was important because it gave rise to the relatively late Neo-Kantianism of (...) Wrocław which was developed more extensively — after Freudenthal’s death — thanks to Richard Honigswald. Neo-Kantianism opposed the prevailing Kantian idealism and experimental psychology at the University of Wrocław. (shrink)