Reductionism provably fails in mathematics. Can it succeed in science?

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.