Two approaches to the logical-mathematical relations: Frege and Schröder (Q2849099)

This paper compares the logical systems of Frege and Schröder, aiming to discover not only dissimilarities, but also similarities between them. The author unfolds his mind in three sections: the first one devoted to Frege's \textit{logicism} (pp.~220--229), the second one to Schröder's work (pp.~229--247), and the third, entitled \textit{Frege and Schröder: different programs with an ideal in common} (pp.~247--249), being a summing up. Yet from the length of these ``sections'' it is clear that the author aimed to sketch the work of Schröder, taking for granted that the reader is familiar with Frege's logicism, and leaving only two pages for the comparison Frege/Schröder. This is not absolutely incidental. The author presupposes the non-knowledge of Schröder and in this he is fully right. Schröder has in the logical literature not the space he deserves.NEWLINENEWLINE According to the author, all work of Schröder revolves around his booklet [\textit{E. Schröder}, Ueber die formalen Elemente der absoluten Algebra. Baden-Baden, Stuttgart: Schweizerbart (1873; JFM 06.0055.01)] in which he introduces a general and formal theory (nowadays we would say ``syntactical''), made up of symbols without meaning, of which logic is only a possible interpretation (nowadays, a ``model''). Then logic makes its appearance as the more general and formal theory and at the same time as a possible semantics. This is clear in \textit{E. Schröder}'s [Vorlesungen über die Algebra der Logik. III. Band. Leipzig: Teubner (1895; JFM 26.0074.01)], where he develops an algebra of relatives, postponing the treatment of the logic of relatives to another book. Unfortunately, with the death of Schröder in 1902, this project was aborted. We have only the algebra of relatives.NEWLINENEWLINE This algebra of relatives enhances the previous 1873 absolute algebra with the introduction of relatives as ordered couples and with the operations obtained among them. Like in the 1873 booklet, logic is only an interpretation. Nevertheless, in this period Schröder attached another meaning to his algebra of relatives: that of being a general language (called \textit{Pasigraphy}) in which the main concepts of the exact sciences could be translated. But this is not all: with the passing of the time, according to the author, Schröder was more and more captured by the allure of logic, until he considered it not only as a \textit{possible interpretation}, but as the \textit{canonical interpretation}; i.e., the privileged model of the formal theory of relatives.NEWLINENEWLINE After much controversy, Schröder espoused a sort of logicism similar in many features to the Fregean one, the difference lying in this: while Frege's \textit{conceptual notation} has an \textit{internal} semantics, Schröder's formal theory has an \textit{external} one. While for Frege there is only one possibile interpretation of this calculus, Schröder's calculus, albeit leaving room for diverse interpretations, has \textit{a posteriori} a canonical interpretation. In other words, if \textit{de jure} for Schröder the algebra of relatives is lacking any meaning, \textit{de facto} it has only one model. The only model deserving to be taken into consideration being the logic of relatives.NEWLINENEWLINE {Caveat}: The reviewer is a scholar engaged in the figure of the mathematician Ernst Schröder. As a matter of fact, he does not share the interpretation of Schröder's work made in this article, but, thinking Zentralblatt MATH is not the place to argue out controversies, he preferred to convey without glosses the substance of this paper to the reader.NEWLINENEWLINEFor the entire collection see [Zbl 1236.03004].