There is a story that in 1930 the great mathematician John von Neumann emerged from a seminar delivered by Kurt Gödel saying: "It's all over." Gödel had just proved the two theorems about the logical foundations of mathematics that are the subject of this valuable new book by Francesco Berto.
At the turn of the century, David Hilbert had challenged mathematicians to prove that mathematics is consistent - it is not possible to prove both of two contradictory results - and had expressed his confidence that mathematics is complete. In other words, it is possible to find the answer to any mathematical problem if one is sufficiently inspired.
Gödel's incompleteness theorems show that if any formal logical system that includes arithmetic is consistent, then it is not complete - there are statements that cannot be proved or disproved in the system - and that such a system, if consistent, cannot prove its own consistency. For some, these are damaging to the integrity of mathematics. For others, they seem to provide the perfect riposte to any philosopher demanding solid foundations for mathematics: since Gödel has shown we can't provide them, we can just ignore anyone with such quibbles and carry on proving theorems!
It might have been expected that Gödel's theorems, which appeared to destroy the hopes of 20th-century mathematicians that their subject might be placed on solid foundations, would change the practice of mathematics. But in practice most mathematicians (including von Neumann) have continued to work much as before.
On the other hand, Gödel's results have been used extensively, if not always appropriately, by many outside mathematics. Postmodern philosophers have drawn interesting conclusions from them; Roger Penrose has based on them his argument that human intelligence is intrinsically different from that of computers; and in 1979, Douglas Hofstadter wrote his wonderfully idiosyncratic and infuriating Gödel, Escher, Bach: An Eternal Golden Braid, which brought to many (including myself and Berto) the excitement of Gödelian logic.
In the book's first half, Berto leads the reader through the proof of Gödel's theorems, which he calls the "Gödelian Symphony". This is inevitably technical and quite demanding, but Berto's clear exposition and his strategy of dividing the proof into short, easily digestible chunks make it pleasant reading (and the use of footnotes rather than endnotes greatly helps the reader).
He then looks at the uses to which the theorems have been put. Here Berto is lucid and witty in exposing mistaken applications of Gödel's results to systems that don't meet the conditions necessary for the proofs to work, while nevertheless acknowledging that invalid applications can still be of interest as metaphors.
His account of Penrose's arguments about artificial intelligence is equally clear and illuminating.
But the heart of the book, for me as a mathematician, lies in the chapters on the philosophy of mathematics. Berto explains how Gödel remained a Platonist in his view of mathematics and shows that such a view is consistent with the incompleteness theorems.
In his final section, he discusses the case of Ludwig Wittgenstein, whose response to Gödel has seemed puzzling. Here he shows that some recent ideas in logic seem to make Wittgenstein's position more plausible than it has sometimes seemed. This discussion of paraconsistent logical systems, in which some statements can simultaneously be both true and false, but which avoid the disastrous consequence that all propositions are true within the system, provides a stimulating conclusion to this fascinating book.
Berto has provided a thoroughly recommendable guide to Gödel's theorems and their current status within, and outside, mathematical logic.
There's Something About Gödel: The Complete Guide to the Incompleteness Theorem
By Francesco Berto. Wiley-Blackwell. 256pp, £50.00 and £14.99. ISBN 9781405197663 and 97670. Published 6 November 2009