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.