The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of
Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and prooftheoretic programme. In this period, there were also lively exchanges between the various schools culminating in the famous Hilbert-Brouwer controversy in the 1920s. The state of the foundational programmes at the end of the classical period is reported in the papers by
Carnap, Heyting and von Neumann (Cf. Benacerraf and Putnam (1983)) from the Conference on Epistemology of the Exact Sciences in Königsberg 1930. This was the very same symposium at which Gödel announced his First Incompleteness Theorem.
In this presentation I will concentrate on various varieties of Logicism inspired by Frege's original version which was outlined informally in his Grundlagen der Arithmetik (1884) and presented in formal detail in his Grundgesetze der Arithmetik (1893/1903). I will discuss the motivation behind the logicist programme and the severe difficulties that it faces. In view of these difficulties it is tempting to pronounce the "Death of Logicism"; and obviously the programme cannot be pursued in anything like its original form. The original goal of showing that substantial mathematical theories can be based conceptually, epistemologically, and
ontologically solely on something that deserves the label “logic” cannot—as far as I can see—be achieved. However, there are close connections between logic and mathematics that should be explored. Much can be learned both philosophically and technically from considering
various logicist and neologicist programmes. Moreover, Logicism very much resembles the boar Särimner that according to Nordic mythology was slaughtered every night and eaten—just to be brought back to life the next day.
2007. , 14 p.14- p.
SCAS (The Swedish Collegium for Advanced Study), Uppsala, Autumn Semester 2007
Detta papper presenterades vid ett seminarium vid SCAS (The Swedish Collegium for Advanced Study), Uppsala, Autumn Semester 2007.