The Fabric of Reality David Deutch

particular ‘abstract computer’ does the job. The proof I have given of the
existence of Cantgotu environments is essentially due to Turing. As I said,
he was not thinking explicitly in terms of virtual reality, but an ‘environment
that can be rendered’ does correspond to a class of mathematical questions
whose answers can be calculated. Those questions are 
computable. The
remainder, the questions for which there is no way of calculating the answer,
are called 
non-computable. If a question is non-computable that does not
mean that it has no answer, or that its answer is in any sense ill-defined or
ambiguous. On the contrary, it means that it definitely has an answer. It is
just that physically there is no way, even in principle, of obtaining that
answer (or more precisely, since one could always make a lucky,
unverifiable guess, of proving that it is the answer). For example, a 
prime
pair is a pair of prime numbers whose difference is 2, such as 3 and 5, or 11
and 13. Mathematicians have tried in vain to answer the question whether
there are infinitely many such pairs, or only a finite number of them. It is not
even known whether this question is computable. Let us suppose that it is
not. That is to say that no one, and no computer, can ever produce a proof
either that there are only finitely many prime pairs or that there are infinitely
many. Even so, the question does have an answer: one can say with
certainty that either there is a highest prime pair or there are infinitely many
prime pairs; there is no third possibility. The question remains well-defined,
even though we may never know the answer.
In virtual-reality terms: no physically possible virtual-reality generator can
render an environment in which answers to non-computable questions are
provided to the user on demand. Such environments are of the Cantgotu
type. And conversely, every Cantgotu environment corresponds to a class of
mathematical questions (‘what would happen next in an environment defined
in such-and-such a way?’) which it is physically impossible to answer.
Although non-computable questions are infinitely more numerous than
computable ones, they tend to be more esoteric. That is no accident. It is
because the parts of mathematics that we tend to consider the least esoteric
are those we see reflected in the behaviour of physical objects in familiar
situations. In such cases we can often use those physical objects to answer
questions about the corresponding mathematical relationships. For example,
we can count on our fingers because the physics of fingers naturally mimics
the arithmetic of the whole numbers from zero to ten.
The repertoires of the three very different abstract computers defined by
Turing, Church and Post were soon proved to be identical. So have the
repertoires of all abstract models of mathematical computation that have
since been proposed. This is deemed to lend support to the Church-Turing
conjecture and to the universality of the universal Turing machine. However,
the computing power of 
abstract machines has no bearing on what is

---

computable in reality. The scope of virtual reality, and its wider implications
for the comprehensibility of nature and other aspects of the fabric of reality,
depends on whether the relevant computers are physically realizable. In