Frege's conception of logic (Q2894726)

Part of the mind of great innovators often lives in a distant past, and their work contains not only revolutionary breakthroughs opening up unimagined vistas, but also, as if to counterbalance the daringness of their vision, archaic elements of concreteness. Copernicus retained the celestial spheres, for Pasch geometry was ``a branch of natural science'' and ``an empirical science'', and Frege, the inventor of the symbolic language of modern logic, believed that axioms cannot be chosen arbitrarily, as they have to be ``true'' (in an absolute way), turning logic in the mathematical context back to a time that perhaps -- if \textit{I. Tóth} [Fragmente und Spuren nichteuklidischer Geometrie bei Aristoteles. Beiträge zur Altertumskunde 280. Berlin: de Gruyter (2010; Zbl 1219.01005)] or \textit{V. Hösle} [I fondamenti dell'aritmetica e della geometria di Platone. Milano: Vita e Pensiero (1994)] are right -- even precedes Plato and Aristotle.NEWLINENEWLINEFrege's correspondence with Hilbert on the subject of Hilbert's axiomatic approach in \textit{Grundlagen der Geometrie}, carried between December 1899 and September 1900, which shows his thorough unwillingness to accept Hilbert's approach, an unwillingness that is espoused in public in [\textit{G. Frege}, Deutsche Math.-Ver. 12, 319--324, 368--375 (1903; JFM 34.0525.02)] as well, greatly diminished Frege's standing with mathematicians. Even before the onset of the ``Frege industry'' in philosophical circles (Frege being considered the father of analytic philosophy, still dominant in the English-speaking world), \textit{H. Freudenthal} would write in [Logic, methodology and philosophy of science, Proc. 1960 Int. Congr., 613--621 (1962; Zbl 0136.00205)]: ``Frege, rebuking Hilbert like a schoolboy, also joins the Bœtians. (I have never understood why he is so highly esteemed today)'' (p.\ 618).NEWLINENEWLINEThe aim of Blanchette's book is to explain, in as clear terms as possible, why Frege's understanding of logic, although repudiated by modern logic in both its proof-theoretical and its model-theoretical approaches, is coherent and defensible when applied in a common language context. The key concept that is seen as determining Frege's conception of logic is that of ``thoughts''. Logical sentences are not to be seen as formal expressions only, as they express ``thoughts'', have ``conceptual content''; Frege does not allow for non-interpreted primitive notions. His logicism, the reduction of arithmetic to logic is presented in a very lucid manner, emphasizing the role ``conceptual analysis'' has in turning a sentence of ordinary arithmetic into one of the formal language, as is the Frege-Hilbert dispute on the nature of the axioms of geometry. Frege is shown to not only oppose the use of models to prove independence or consistency, as these allow the free interpretation of the primitive notions, but would not recognize even a purely syntactic proof of the non-derivability of a sentence \(\varphi\) from a certain axiom system \(\Sigma\) (such as the proofs pioneered by \textit{T. Skolem} [Krist. Vid. Selsk. Skr. I, 1920, Nr. 4, 36 S. (1922; JFM 48.1121.01)]) as a proof of \(\varphi\)'s independence from \(\Sigma\), as the set of ``thoughts'' expressed by \(\Sigma\) may well imply the ``thought'' expressed by \(\varphi\). A separate chapter looks in greater detail at Frege's views on semantics and models, and another one is devoted to the few instances in which Frege presents elements of a metatheory, and why there is hardly anything on soundness, completeness, consistency in Frege's work.NEWLINENEWLINEThe author defends Frege in a great many instances, and passes in silence the complete absence of a counter-proposal for how an axiomatics of geometry would have to look like to be acceptable to his conception of logic (Frege advances only the thesis that arithmetic can be reduced to logic), nor is she critical of a passage from Frege's notes to Jourdain of 1910 she quotes on p.\ 128 (``The indemonstrability of the axiom of parallels cannot be proven''). Thinking largely of the common language context, with fixed meanings (references) attributed to words, the author concludes that ``despite the convenience of the modern approach [\(\ldots\)] nevertheless Frege was right: the syntactic consistency of a set, though an important feature in its own right, is no guarantee of the consistency, in the ordinary sense, of what's expressed by the members of that set.'' (p.\ 178)