THE FREGE-HILBERT CONTROVERSY

FREGE, G.

Über die Grundlagen der Geometrie + Über die Grundlagen der Geometrie II.

Leipzig, B. G. Teubner, 1903. 8vo. Bound with the original wrappers in contemporary half cloth with gilt lettering to spine. In "Jahresbericht der Deutschen Mathematiker-Vereinigung", 12. Band. 6. Heft. Juni. Bound with all issues from December 1902 till December 1903 (all issues with wrappers). Pp. 319-324; Pp. 368-375. [Entire volume: VI, 602 pp.].

First appearance of Frege's important paper on the role of axioms in mathematical theories, describing the correct way to demonstrate consistency and independence results for such axioms.
The two papers was Frege's response to Hilbert's "Grundlagen der Geometrie" which inaugurated the famous Frege-Hilbert Controversy.

"Hilbert's lecture [Grundlagen der Geometrie] inspired a sharp reaction from his contemporary Gottlob Frege, who found both Hilbert's understanding of axioms, and his approach to consistency and independence demonstrations, virtually incomprehensible and at any rate seriously flawed. Frege's reaction is first laid out in his correspondence with Hilbert from December 1899 to September 1900, and subsequently in two series of essays (both entitled "On the Foundations of Geometry") published in 1903 and 1906. Hilbert was never moved by Frege's criticisms, and did not respond to them after 1900. Frege, for his part, was never convinced of the reliability of Hilbert's methods, and held until the end that the latter's consistency and independence proofs were fatally flawed."(SEP).

"Frege felt that his view represented a traditional understanding of this notion, and that Hilbert's departure from this understanding led to a confusion about axioms that undermined many of the sorts of results, in particular, the independence of the axioms of geometry, that Hilbert saw as major mathematical achievements. Symptomatic of Hilbert's confusion, according to Frege, was Hilbert's claiming that axioms could serve to define; the reason that this is a confusion, according to Frege, is that axioms and definitions are statements of wholly different types." (Antonelli, Frege's New Science).

Friedrich Ludwig Gottlob Frege (1848 - 1925) was a German mathematician, but his main contributions lie in his becoming a logician and a philosopher, who influenced the fields of logic and analytic philosophy immensely. Together with Wittgenstein, Russel and Moore, Frege is considered the founder of analytic philosophy, and a main founder of modern mathematical logic. In the preface of the "Principia Mathematica" Russell and Whitehead state that "In all questions of logical analysis our chief debt is to Frege" (p. VIII). His influence on 20th century philosophy has been deeply profound, especially in the English speaking countries from the middle of the 20th century and onwards; in this period most of his works were translated into English for the first time.
The philosophical papers of Frege were published in Germany in scholarly journals, which were barely read outside of German speaking countries. The first collections of his writings did not appear until after the Second World War, and Frege was little known as a philosopher during his lifetime. He greatly influenced the likes of Russel, wittgenstein and Carnap, though, and bears a great responsibility for the turn modern philosophical thought has taken. Due to his contributions to the philosophy of language, analytic philosophy could be founded as it were.
Instead of answering the question about meaning, Frege here sets out to explore the foundations of arithmetic, beginning with questions such as "What is a number?" In his solutions the answer to the question of meaning could also be found, though, and he permitted himself "the hope that even the philosophers, if they examine what I have written without prejudice, will find in it something of use to them." (p. XIi - Introduction).

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