This is a collection of 21 expanded contributions to Gödel’s centenary celebration at the University of Vienna, in 2006. Takeuti, former president of the Kurt Gödel Society, writes in the foreword: “For everyone who gathered in his honor, Gödel’s extraordinary contributions to and tremendous influence in mathematics would be something of which we were already deeply aware.” Gödel has been thought by many to be the greatest logician since Aristotle; the impact of his work can be seen not only in mathematics and logic, but also in philosophy, physics, and, more generally, systems thinking. To quote Manin, a great mathematician:

Some profound thinkers [...] express[ed] the hope that there should exist several universal laws from which all other truths could be deduced by pure reason. After Gödel’s work, however, we can be sure that all of these hopes are unfounded. By pure deductive reasoning, one cannot even deduce from a finite number of basic principles all true statements about integers that can be formulated in the language of high school algebra. [1]

Anyone looking for an accessible introduction to the conceptual aspects of Gödel’s theorem and its proof should read Manin’s essay.

The book is divided into three parts: “Historical Context,” “A Wider Vision: The Interdisciplinary, Philosophical and Theological Implications of Gödel’s Work,” and “New Frontiers: Beyond Gödel’s Work in Mathematics and Symbolic Logic.” Some papers are very technical and specialized, while others are of interest to a wider audience of “educated and informed others.” Certain authors--such as Angus Macintyre--claim that Gödel’s results had little effect on “current [pure] mathematics,” while others show that these results have had a substantial influence on computing--for example, Jack Copeland observes that “with Gödel’s introduction of the idea of representing logical and arithmetical statements as numbers, together with his foundational contributions to recursion theory, the new machines might [...] be described [...] as ‘Gödel in hardware.’” The creative character of mathematics--illustrated by Carnap’s observation on his conversation with Gödel in 1929: “Mathematics is inexhaustible: one must always again draw afresh from the fountain of intuition”--is stressed by Juliette Kennedy in her nice paper on the appreciation of Gödel’s thesis. This is in excellent agreement with Manin’s reformulation of Gödel’s theorem: “In order to generate all true statements about integers, one needs infinitely many new ideas” [1].

Georg Kreisel, in his rather technical paper, observes that it is an open secret that logical deductions are not a particularly demanding or rewarding part of proofs compared with spotting suitable ideas. Some of us may remember E. W. Dijkstra’s clear distinction between two kinds of thinking: (informal) pondering and (formal) reasoning [2]. Kreisel also notes that mathematicians expect from a proof not only truth but also understanding. Grattan-Guinness’ more accessible and lucid paper summarizes the significance of Gödel’s theorem and of its proof method for mathematics and logic, and then, among other things, refers to the role of James R. Newman--the compiler of the four-volume set The world of mathematics [3] (the set includes Gödel’s results)--as a great popularizer.

Sigmund’s very readable paper on Gödel’s Vienna years includes many documents revealing the “everyday chicanery” of life in Vienna and at the University of Vienna immediately before and after the Anschluss. The fascinating paper by Solomon Feferman overviews the lengthy correspondence between Gödel and Hilbert’s assistant and junior colleague Bernays, and concentrates on Gödel’s views on finitism, constructivity, and Hilbert’s program.

Papadimitriou clearly explains both the role of Gödel’s results and the need to define computation: in order to show that something is impossible, “one must first define and chart with precision the whole realm of possibilities.” He notes the importance of the fact that four independent and radically different definitions of computation (by Turing, Church, Kleene, and Post) are equivalent, and observes that the “real-world incarnation” of the mathematical concept of computation “would not be very long in coming.” The following paper by Jack Copeland provides an outstanding overview of the early history of computers used for cryptanalysis in the UK during World War II--years earlier than the ENIAC. Copeland also observes that Turing and Wilkinson, in the late 1940s, prepared a large library of routines for the not-yet-existent machine, in the same manner as described later by E. W. Dijkstra. Turing’s role in the design and development of hardware, which is rarely mentioned in the “inorthodox histories of the computer,” is also clearly delineated. Finally, Copeland presents Gödel’s fundamental quote: “Mind is not mechanical. [...] Mind cannot understand its own mechanism.” This essential property of complex systems--regrettably not emphasized in the book--was suggested by the great systems thinker F. A. Hayek in 1942; formulated by him more fully in 1952 [4], without reference to Gödel; and then again in 1962 [5] with an explicit reference to “the celebrated theorem due to Kurt Gödel.”

Rindler’s paper describes Gödel’s cosmological model universe that “was consistent with general relativity but that nevertheless exhibited two startlingly disturbing features: bulk rotation (but with respect to what, as there is no absolute space in general relativity?) and travel routes into the past (enabling one to witness or even prevent one’s own birth?).” Karl Svocil’s interesting but rather unfocused paper on physical unknowables has no explicit references to Gödel, and the (first several sections of the) much more accessible paper by John Barrow, “Gödel and Physics,” may be used as a roadmap for the less initiated reader.

Gödel’s ontological proof of the existence of God as an embodiment of all positive properties is discussed by Denys Turner in the context of “medieval theologians Thomas Aquinas, Meister Eckhart and Nicholas of Cusa [who] conceived of their discipline as being ‘systematically incomplete’ and open to a region beyond its power to determine” and compared with Wittgenstein’s thesis that propositions underlying the possibility of language can be “shown” but not “said” (within language). Turner observes that Gödel’s contribution to skeptical questionings “exploded the myth of a narrowly rationalist and essentially anti-theological conception of reason.” Pete Hajek presents Gödel’s ontological proof and its variants in a very technical manner. Piergiorgio Odifreddi in his very short paper claims that Gödel’s main results can be interpreted as mathematically precise formulations of intuitions by Aristotle, Leibniz, and Kant and that “(part of) philosophy can be reinterpreted as asking questions and suggesting answers that mathematics makes precise.”

Putnam’s paper targets the implausible Chomskian conjecture that “our total competence can be represented by a Turing machine” and discusses the arguments by John Lucas and Roger Penrose about our minds. Lucas assumes standard physics and concludes that our minds cannot be identical with any material system; Penrose, in his paper, claims that, while the actions of the mind “must transcend computation” (compare, again, with F. A. Hayek’s contributions [4,5]), these noncomputational processes are physical, and a basic change in physical theory is needed to account for them.

Kohlenbach’s paper, “Gödel’s Functional Interpretation and Its Use in Current Mathematics,” and Hugh Woodin’s paper, “The Transfinite Universe,” are very technical and require substantial prerequisites for their understanding. Friedman’s technical but much more accessible paper surveys various interesting results extending Gödel’s contributions on incompleteness phenomena; quite a few of these results refer to Friedman’s unpublished papers available online. The short essay by Cohen on the origin and background of his famous result showing the independence of the continuum hypothesis from the usual set theory axioms and on his professional relationships with Gödel is of great interest.

The final excellent paper by Wigderson is about interesting but unknowable mathematical truths if knowability is “interpreted by modern standards, namely, via computational complexity.” Wigderson includes the translation of Gödel’s famous letter to von Neumann, written in 1956, and argues that a real understanding of a mathematical structure requires an efficient algorithm rather than just a decision procedure. He shows that nondeterministic polynomial-time (NP) completeness results providing “the stamp of difficulty” are ubiquitous in science and mathematics, and concludes that they “may perhaps better capture Gödel’s legacy of what is unknowable in mathematics.”

In summary, the book presents an interesting but uneven collection of papers. Some excellent papers would be very worth reading and are accessible to a wide audience; other papers are overly specialized. In addition, the index is incomplete (for example, Carnap and Wittgenstein are omitted).