From John Lennox in his book God's Undertaker: Has Science Buried God? (pp 52-5):
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 Gödel 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 Gödel 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, Gödel 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 Gödel's First Incompleteness Theorem.
Now Hilbert's Programme also aimed to prove the essential consistency of his formulation of mathematics as a formal system. Gödel, 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, 'Gödel proved that in mathematics the whole is always greater than the sum of the parts'. 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.
That there are limits for reductionism in science itself is borne out by the history of science, which teaches us that it is important to balance our justifiable enthusiasm for reductionism by bearing in mind that there may well be (and usually is) more to a given whole than simply what we obtain by adding up all that we have learned from the parts. Studying all the parts of a watch separately will not necessarily enable you to grasp how the complete watch works as an integrated whole. There is more to water than we can readily see by investigating separately the hydrogen and oxygen of which it is composed. There are many composite systems in which understanding the individual parts of the system may well be simply impossible without an understanding of the system as a whole – the living cell, for instance.
Besides methodological reductionism, there are two further important types of reductionism: epistemological and ontological. Epistemological reductionism is the view that higher level phenomena can be explained by processes at a lower level. The strong epistemological reductionist thesis is that such 'bottom-up' explanations can always be achieved without remainder. That is, chemistry can ultimately be explained by physics; biochemistry by chemistry; biology by biochemistry; psychology by biology; sociology by brain science; and theology by sociology. As the Nobel Prize-winning molecular biologist Francis Crick puts it: The ultimate aim of the modern development in biology is, in fact, to explain all biology in terms of physics and chemistry.'
This view is shared by Richard Dawkins. 'My task is to explain elephants, and the world of complex things, in terms of the simple things that physicists either understand, or are working on.' Leaving aside for the moment the very questionable assertion to which we must return below that the subject matter of physics is simple (think of quantum mechanics, quantum electrodynamics or string theory), the ultimate goal of such reductionism is evidently to reduce all human behaviour – our likes and dislikes, the entire mental landscape of our lives – to physics. This view is often called 'physicalism', a particularly strong form of materialism. It is not, however, a view which commends universal support, and that for very good reasons. As Karl Popper points out: 'There is almost always an unresolved residue left by even the most successful attempts at reduction.'
Scientist and philosopher Michael Polanyi helps us see why it is intrinsically implausible to expect epistemological reductionism to work in every circumstance. He asks us to think of the various levels of process involved in constructing an office building with bricks. First of all there is the process of extracting the raw materials out of which the bricks have to be made. Then there are the successively higher levels of making the bricks – they do not make themselves; brick-laying – the bricks do not 'self-assemble'; designing the building – it does not design itself; and planning the town in which the building is to be built – it does not organize itself. Each level has its own rules. The laws of physics and chemistry govern the raw material of the bricks; technology prescribes the art of brick-making; brick-layers lay the bricks as directed by the builders; architecture teaches the builders; and the architects are controlled by the town planners. Each level is controlled by the level above. But the reverse is not true. The laws of a higher level cannot be derived from the laws of a lower level – although what can be done at a higher level will, of course, depend on the lower levels. For example, if the bricks are not strong there will be a limit on the height of the building that can safely be built with them.
Or take another example, quite literally to your hand at this moment. Consider the page you are reading just now. It consists of paper imprinted with ink (or perhaps it is a series of dots on the computer screen in front of you). It is surely obvious that the physics and chemistry of ink and paper (or pixels on a computer monitor) can never, even in principle, tell you anything about the significance of the shapes of the letters on the page; and this has nothing to do with the fact that physics and chemistry are not yet sufficiently advanced to deal with this question. Even if we allow these sciences another 1,000 years of development it will make no difference, because the shapes of those letters demand a totally new and higher level of explanation than physics and chemistry are capable of giving. In fact, complete explanation can only be given in terms of the higher level concepts of language and authorship, the communication of a message by a person. The ink and paper are carriers of the message, but the message certainly does not arise automatically from them. Furthermore, when it comes to language itself, there is again a sequence of levels. You cannot derive a vocabulary from phonetics, or the grammar of a language from its vocabulary, etc.
As is well known, the genetic material DNA carries information. We shall describe this later on in some detail; but the basic idea is that DNA can be thought of as a long tape on which there is a string of letters written in a four-letter chemical language. The sequence of letters contains coded instructions (information) that the cell uses to make proteins. But the order of the sequence is not generated by the chemistry of the base letters.
In each of the situations described above, we have a series of levels, each higher than the previous one. What happens on a higher level is not completely derivable from what happens on the level beneath it. In this situation it is sometimes said that the higher level phenomena 'emerge' from the lower level. Unfortunately, however, the word 'emerge' is easily misunderstood, and even misleadingly misused, to mean that the higher level properties arise automatically from the lower level properties without any further input of information or organization – just as the higher level properties of water emerge from combining oxygen and hydrogen. However, this is clearly false in general, as we showed earlier by considering building and writing on paper. The building does not emerge from the bricks nor the writing from the paper and ink without the injection of both energy and intelligent activity.