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There is no paradox concerning the barber (‘Shaking the foundations
of mathematics’, 21 November). Even if John Barrow had formulated the pseudo-paradox
correctly, all it would have shown is that no village could exist in which
the resident barber shaves all and only those residents who do not shave
themselves. Moreover, while positing a ‘set of all sets’ does lead to a
paradox, this requires a more sophisticated argument than to ask: is it
a member of itself? Barrow was apparently thinking of the ‘Russell Set’:
the set of all sets which are not members of themselves. This ‘set’, which
would have to be a member of itself if and only if it were not a member
of itself, clearly cannot exist, although its existence can be established
in ‘naive’ (Cantorian) set theory – which is therefore inconsistent.

Contrary to Barrow’s assertion, reductio ad absurdum is not rejected
by constructivists, but especially beloved of them. In fact, for intuitionists,
we are justified in asserting not-P precisely when we are able to deduce
(or to show how to deduce) an absurdity from P. This is not to say that,
on deducing an absurdity from not-P, we are entitled to assert P. We are
only entitled to assert not-not-P. Intuitionists reject one half of the
classical law of double negation. Specifically, while they accept the inference
from P to not-not-P, they disallow the converse.

Richard Benham University of Adelaide Australia

* * *

John Barrow replies: With regard to logical ‘paradoxes’, you miss the
point. My purpose was just to flag them by their titles to make the point
that they were perceived as creating a problem for mathematics and logic
and this played a role in the quest for the foundations and meaning of mathematics
around the turn of the century. Whether or not they really are paradoxes
is another question which, although interesting, was not one I sought to
raise. With regard to intuitionism, I think you are right to object to
my oversimplified popularisation of the reductio ad absurdum. Perhaps I
can correct this for readers. There are two versions of the argument used
in ordinary logic: the first assumes ‘something’ to be false and arrives
at a contradiction and so concludes ‘something’ was true (for example, Euclid’s
proof that the square root of two is an irrational number); the second assumes
‘something’ to be true and hence derives a contradiction, so ‘something’
was false (for example, Euclid’s proof that the number of primes is not
finite). Constructivism in the hands of the intuitionists allows the second
but not the first of these devices. Cosmological singularity theorems fall
into the second category, but proofs that black holes cannot break up or
some formulations of the famous ‘cosmic censorship’ hypothesis fall into
the first.

This correspondence is now closed – Ed.