From John Lennox's God's Undertaker, 52-53.
The
great mathematician David Hilbert, spurred on by the singular
achievements
of mathematical compression, thought that the reductionist
programme
of mathematics could be carried out to such an extent that in
the
end all of mathematics could be compressed into a collection of formal
statements
in a finite set of symbols together with a finite set of axioms and
rules
of inference. It was a seductive thought with the ultimate in ‘bottom-up’
explanation
as the glittering prize. Mathematics, if Hilbert’s Programme
were
to succeed, would henceforth be reduced to a set of written marks
that
could be manipulated according to prescribed rules without any
attention
being paid to the applications that would give ‘significance’ to
those
marks. In particular, the truth or falsity of any given string of symbols
would
be decided by some general algorithmic process. The hunt was
on
to solve the so-called Entscheidungsproblem by finding that general
decision
procedure.
Experience
suggested to Hilbert and others that the Entscheidungsproblem
would
be solved positively. But their intuition proved wrong. In 1931
the
Austrian mathematician Kurt Godel published a paper entitled ‘On
Formally
Undecidable Propositions of Principia Mathematica and Related
Systems’.
His paper, though only twenty-five pages long, caused the
mathematical
equivalent of an earthquake whose reverberations are still
palpable.
For Godel had actually proved that Hilbert’s Programme was
doomed
in that it was unrealizable. In a piece of mathematics that stands
as
an intellectual tour-de-force of the first magnitude, Godel demonstrated
that
the arithmetic with which we are all familiar is incomplete: that is,
in
any system that has a finite set of axioms and rules of inference and
which
is large enough to contain ordinary arithmetic, there are always true
statements
of the system that cannot be proved on the basis of that set of
axioms
and those rules of inference. This result is known as Godel’s First
Incompleteness
Theorem.
Now
Hilbert’s Programme also aimed to prove the essential consistency
of
his formulation of mathematics as a formal system. Godel, in his
Second
Incompleteness Theorem, shattered that hope as well. He proved
that
one of the statements that cannot be proved in a sufficiently strong
formal
system is the consistency of the system itself. In other words, if
arithmetic
is consistent then that fact is one of the things that cannot be
proved
in the system. It is something that we can only believe on the basis
of
the evidence, or by appeal to higher axioms. This has been succinctly
summarized
by saying that if a religion is something whose foundations
are
based on faith, then mathematics is the only religion that can prove it
is
a religion!
In
informal terms, as the British-born American physicist and
mathematician
Freeman Dyson puts it, ‘Godel proved that in mathematics
the
whole is always greater than the sum of the parts’.10 Thus there is a limit
to
reductionism. Therefore, Peter Atkins’ statement, cited earlier, that ‘the
only
grounds for supposing that reductionism will fail are pessimism in
the
minds of the scientists and fear in the minds of the religious’ is simply
incorrect.
