Kurt Gödel: a mathematical myth. (Kurt Gödel: ein mathematischer Mythos.) (German) Zbl 0872.01030

DePauli-Schimanovich, Werner; Weibel, Peter

Wien: Hölder-Pichler-Tempsky. 146 S. (1997).

The book is based on the script of a film with the same title which was telecast in December 1986 in the Austrian Television (ORF). The aim of this book is to convey the fascination of mathematics with the help of the biography and the work of the fascinating genius Kurt Gödel. Mathematics is shown as being “our invisible culture”, as an enterprise at the intersections between philosophy, physics and computer science, influencing branches like philosophy of science, artificial intelligence, cognitive science and logic.
All these branches are connected in the opening section on the myth of Kurt Gödel. The next five sections deal with Gödel’s biography, with a vivid description of his life in the intellectual melting pot of Vienna in the period between the two World Wars. Sections 7 and 8 deal with the consequences of Gödel’s results for computer science, especially the impact of the incompleteness theorem on the modern dream of mechanized reasoning. The authors discuss the correspondence between formal proof systems in logic and computability which is illustrated with the help of Turing machines. They relate recursiveness to Turing computability. In section 8 the authors exemplify Church’s undecidability problem being implied by Gödel’s incompleteness theorem with the halting problem of Turing machines, i.e. the question whether there is a universal Turing machine which is able to prove for any input and for any Turing machine that the machine will halt computation. The authors prove the unsolvability of this problem using graphical illustrations developed for the film. In section 9 the authors relate Gödel’s results to the foundational crisis in mathematics. Section 10 describes Gödel’s thoughts in cosmology, dealing with rotating universes and the theoretical possibility of time journeys into the past. The book ends with a prospect on artificial intelligence.
The authors have reached their aim to illustrate the fascination of logic and mathematics for a non-professional audience. Their success is supported by a very attractive appearance of the book. A lot of rare (but rather small) photographs from Gödel’s life are printed in the text. The technical parts contain drawings and other graphics, the latter being obviously screenshots from the film (and therefore of minor quality).

Reviewer: V.Peckhaus (Erlangen)

MSC:

01A70	Biographies, obituaries, personalia, bibliographies
01A60	History of mathematics in the 20th century

Keywords:

undecidability; incompleteness; Turing machines; halting problem; foundations, proof theory; Hilbert’s program; cosmology; time journeys; philosophy; physics; computer science; artificial intelligence; cognitive science; logic; Turing computability; undecidability problem