Book Review of: FRANCESCO BERTO. There's Something
about Godel. Malden, Mass., and Oxford: Wiley-Blackwell,
2009. ISBN 978-1-4051-9766-3 (hbk); 978-1-4051-9767-0
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Citation
McGee, V. “Book Review of: FRANCESCO BERTO. There’s
Something About Godel. Malden, Mass., and Oxford: WileyBlackwell, 2009. ISBN 978-1-4051-9766-3 (hbk); 978-1-40519767-0 (pbk). Pp. Xx + 233. English Translation of Tutti Pazzi
Per Godel! (Rome: Gius, Laterza & Figli, 2008).” Philosophia
Mathematica 19, no. 3 (August 23, 2011): 367–369.
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Francesco Berto. There’s Something about Gödel. Malden, Mass., and Oxford: Wiley-Blackwell,
2009. ISBN 978-1-4051-9766-3 (cloth) and 978-1-4051-9767-0 (paper). Pp. xx + 233. English
translation of Tutti pazzi per Gödel! (Rome: Gius, Laterza & Figli, 2008).
There’s Something about Gödel is a bargain: two books in one. The first half is a gentle
but rigorous introduction to the incompleteness theorems for the mathematically uninitiated. The
second is a survey of the philosophical, psychological, and sociological consequences people
have attempted to derive from the theorems, some of them quite fantastical.
The first part, which stays close to Gödel’s original proofs, strikes a nice balance, giving
enough details that the reader understands what’s going in the in proofs, without giving so many
that the reader feels overburdened. Perhaps he skimps too much on details, as when he decides
not to explain how to convert recursive definitions into explicit ones. Also, I wish he had talked
about Löb’s theorem. But these are small complaints.
The second half discusses a sampling of what one reads about Gödel’s theorems in
philosophy journals and in the popular press, and here Berto often finds himself exasperated,
especially by the postmodernists. One only has to set what Gödel proved alongside what the
postmodernist philosophers say Gödel proved to share his sense of bewilderment.
A more engaged discussion assesses the connection of the incompleteness theorems to
Platonism, which Berto understands as the belief in a mythical intended model that we know
about by some mysterious faculty of intuition. The semantic version of the first incompleteness
theorem, which says that if Peano arithmetic (PA) is consistent, then its Gödel sentence – call it
“γ” – is true, commits us to Platonism, Berto says, because it requires the existence of the
mythical intended model, whereas the syntactic version, which says (in Rosser’s improved
formulation) that, if PA is consistent, it is incomplete, does not. I suspect that this overestimates
what the semantic version of the theorem requires. PA implies “(CON(PA) → γ),” and to get
from this to “(CON(PA) → T(┌ γ ┐),” we only need the T-sentence, “(T(┌ γ ┐) ↔ γ).” Tarski
[1944] has argued forcefully that a mathematical theory of truth that implies the T-sentences
(formulated in a metalanguage) can be neutral with respect to issues of metaphysics and
epistemology. I also suspect that Berto underestimates the presumptions of the syntactic version
of the theorem. A syntactic theory capable of talking about proofs that could be constructed, but
perhaps never are, out of sentences many of which are never actually written down, will need, it
seems, to talk about expression types, and not just tokens, and the rudimentary syntax of
expression types is bi-interpretable, as Quine [1946] shows, with arithmetic.
There is a judicious discussion of the Lucas-Penrose argument that the incompleteness
theorems show that the human mind cannot be simulated by a Turing machine. The presentation
is careless at points. We read [p. 180] that the argument requires that, “for any machine,... we can
always ‘see’ the truth of the relevant Gödel sentence.” The argument doesn’t demand so much; it
only requires that we see the truth of the Gödel sentence for the particular machine (assuming,
for reductio, that there is one) that mimics us. Despite occasional lapses, however, the discussion
is helpful.
Berto is careful to separate the consequences of Gödel’s theorems within the sensible
confines of ordinary mathematics, and the deep insights we get from them, from the morals that
have been drawn from the theorems in philosophy and popular culture, which are often wild and
fanciful, like the idea that Gödel showed the inadequacy of scripture as a source of revealed
truth. In the very last chapter, however, he seems to have joined the other side. He wants to
uphold two ideas of Wittgenstein [1978]: first, that a contradiction in a formal system needn’t be
such a bad thing, and second, that, with regard to a formal mathematical system, there is no
usable notion of truth that reaches beyond provability. The intuitionists also identify truth with
provability, although for them provability isn’t confined to a formal system, but Berto wants to
uphold the identification while maintaining classical reductio ad absurdum. He proposes to do
this by adopting a paraconsistent system that yields all the truths acknowledged by classical
arithmetic, together with some additional, nonclassical truths, so that, even though there isn’t a
largest number, there is a largest number. That it’s possible to maintain these theses without a
complete collapse, in which every arithmetical sentence is regarded as true, is a remarkable fact
(discovered by Graham Priest [1994]),well worth studying. But it lies well outside the sensible
confines of ordinary mathematics.
References.
Lucas, J. R. “Minds, Machines, and Gödel.” Philosophy 36 (1961): 112-27.
Penrose, Roger. The Emperor’s New Mind. Oxford: Oxford University Press, 1989.
Penrose, Roger. Shadows of the Mind. Oxford: Oxford University Press, 1994.
Priest, Graham. “Is Arithmetic Consistent?” Mind 104 (1994): 337-49.
Quine, W. V. “Concatenation as a Basis for Arithmetic.” Journal of Symbolic Logic 11(1946):
105-114.
Tarski, Alfred. “The Semantic Conception of Truth.” Philosophy and Phenomenological
Research 4 (1944): 1-13.
Wittgenstein, Ludwig. Remarks on the Foundations of Mathematics, revised ed. Cambridge,
Mass., and London: MIT Press, 1978.