FOM: David Deutsch's "The Fabric of Reality"

[Andrej Bauer]

In his popular book "The Fabric of Reality" David Deutsch discusses
various things related to Life, Universe and Everything.

He has a section on mathematics in which he states that:


1. Goedel showed that Hilbert's TENTH problem was insoluble.
   This is on page 234 of my 1997 paperback edition:

     ``Hilbert was to be definitively disappointed. Thirty-one years
     later, Kurt Goedel revolutionized proof theory with a
     root-and-branch refutation from which the mathematical and
     philosophical worlds are still reeling: he proved that Hilbert's
     tenth problem is insoluble.''

   I suppose this could be looked at just as a minor mistake, but it
   certainly does not help convince me that Deutsch knows much about
   the subject he is writing chapters about.

   [In case you are wondering what is wrong: Hilbert's SECOND problem
   is about consistency of arithmetic. The TENTH problem was solved
   in the early 70's and is about Diophantine equations.]


2. On pages 231 and 232 he states that intuitionists, with Brouwer
   being the prime example, deny the existence of infinitely many natural
   numbers:

     ``To this end, the Dutch mathematician Luitzen Egbertus Jan
     Brouwer advocated an extreme conservative strategy for proof
     theory, known as *intuitionism*, which still has adherents to
     this day. Intuitionists try to construe `intuition' in the
     narrowest conceivable way, retaining only what they consider to
     be its unchallengeably self-evident aspects. Then they elevate
     mathematical intuition, thus defined, to a status higher even
     that Plato afforded it: they regard it as being prior even to
     pure logic. ....[text omitted]... For instance, intuitionists
     deny that is is possible to have a direct intuition of any
     infinite entity. Therefore they deny that any infinite sets, such
     as the set of all natural numbers, exist at all. The proposition
     `there exist infinitely many natural numbers' they would consider
     self-evidently false.''

   He tops it off with the following passage about the Law of the
   Excluded Middle on page 232:

     ``For instance, if it is indeed false, as intuitionists maintain,
     that there exist infinitely many natural numbers, then we can
     infer that there must be only finitely many of them. How many?
     And then, however many there are, why can we not form an
     intuition of the next natural number above that one?
     Intuitionists would explain this problem away by pointing out
     that the argument I have just given assumes the validity of
     ordinary logic. In particular, it involves inferring, from the
     fact that there are not infinitely many natural numbers, that
     there must be some particular finite number of them. The relevant
     rule of inference is called the *law of the excluded middle*. It
     says that, for any proposition X (such as `there are infinitely
     many natural numbers'), there is no third possibility between X
     being true and its negation ('there are finitely many numbers')
     being true. Intuitionists coolly deny the law of the excluded
     middle.''

   This from the man who proved the existence of a universal quantum
   Turing machine. In view of this passage, confusing Hilbert's 2nd
   and 10th problem is a triviality. I almost feel like becoming an
   intuitionist.


My question to the FOM readers is whether anyone has pointed out to
Deutsch that the above passages are `imprecise', too but it mildly?

Would it do any good to write to Deutsch and explain the difference
between ultrafinitists and intuitionists, and try to explain just how
badly he is misrepresenting intuitionism and the intuitionistic
arguments about the law of the excluded middle? I am disturbed because
his book is widely read and there is even a mailing list devoted to
it, see http://www.TCS.ac/List/forlist.html.

Andrej Bauer
Mittag-Leffler Institute
Stockholm, Sweden
http://andrej.com