This is a book that I would recommend to those theologians who can understand it because they can sharpen their thinking on some aspects of theology. It is also a nice introductory text about some aspects of mathematics. After reading this book I must say that it did nothing to shatter my faith and Christians should not be afraid to read it.

The book sets out to prove mathematically that:

[1] God does not exist

or at the very least

[2] there is almost certainty of the non-existence of God.

I note that nowhere (unless I missed it) does the author give us a clear definition of what is meant by God. The author's main approach seems to be to prove that any type of god he looks at does not exist.

As an amusing aside I mention that one physicist/mathematician proved mathematically that bumble bees cannot fly. The assumption was that the wings of bumble bees support flight by laminar motion. Since most of us have seen bumble bees fly we know that something was wrong about the proof! The assumption of laminar flow was the problem and as far as I recall bumble bees fly by using vortex shedding. So both the assumptions and the mathematics used have to be correct in order to avoid nonsense.

I would point out that there is no universal consensus amongst mathematicians about whether mathematics is a human invention or an independent reality. (For more discussion see Meaning in Mathematics, edited by J Polkinghorne, Oxford University Press, 2011.) The author tells us that he has a viewpoint lying between the "intuitionists" and "formalism" that is informed by "fictionalism". Thus it appears that he views mathematics as being essentially a "human invention" and rejects Platonism (i.e. presumably rejects the "independent reality" mentioned above).

Many aspects of the workings of the universe appear to work by definite laws e.g. gravity/relativity and electromagnetism. These laws were in operation long before scientists discovered them. They happen to match precise mathematical equations. As a physicist that earned his living as an electrical power engineer I would point in particular to the laws governing electromagnetism (Maxwell's equations). Light waves did not need physicists, mathematicians, or engineers to create the equations first before they came into existence. It is those types of facts that makes some lean towards the "independent reality" viewpoint.

I must point out that I was brought up as a Christian but rejected belief in God for about 40 years then started to regain my faith again. I was a pretty hard core atheist i.e. the universe was self-existent, matter organised in the form of the brain was the sole source of consciousness, and there was no life after death. So death was an eternal release from suffering. Although I was in no hurry to die I felt a great sense of peace about the future: after all the only time I have had problems was after I started to exist. If my present faith in God is in fact wrong then I look forward to future non-existence. Meantime my faith makes life more interesting for me and I enjoy fellowship with my Christian friends. If there is no immortal life that is OK by me!

I find no problem with the concept of God being finite since I do not believe that anything that can be assigned a numerical value (e.g. a real number) can be infinite. At minimum all we require is that God has properties/potentials that are "large enough" to satisfy our concept of what God can do or be. If any property is a trillion times larger than the "large enough value" then that value is large enough for most people to be suitably impressed with that aspect of the magnitude of God. There is no need to get bogged down by infinity. Moreover, by not having to worry about infinities one can talk more logically about one aspect of God being greater than some other aspect of God.

The implications of some of the author's arguments would appear to spill over from the non-existence of an infinite God to the non-existence of mathematical entities such as:

"pi" and "e", where "e" is the limit as N tends to infinity of (1+[1/N])^N, and presumably all the irrational numbers. We have no problem with using a finite number to represent "pi" and "e" and the irrational number root(2) when doing everyday calculations - as long as we use a sufficiently large number of significant figures for them to give sufficiently accurate calculation results.

It seems implicit that the author believes in the objective existence of the Problem of Evil. However, he then tells that goodness, liberty, and justice are subjective and have no reality apart from our mental constructions by which we perceive the world. Why should evil be an “objective reality” and goodness be a “subjective reality"? Surely good and evil should both be regarded the same way i.e. both objective or both subjective.

The author argues against Kurt Godel's argument for the existence of God. Now Godel is much more famous for other work in which he proved a theorem called Godel's theorem, In a nutshell Godel's theorem says that: given a finite set of axioms and then using logic there will be questions that one can pose whose truth or falsity cannot be decided using that set of axioms and the application of logic. We may well have a situation where proofs for and proofs against the existence of God cannot settle the question of God's existence. This leaves open the possibility that it is not necessarily illogical to have faith in God. Moreover, for those that have faith in God then Godel's theorem indicates that some truths about God cannot be deduced from axioms and logic: so we rely on what God has revealed to us in sacred books e.g. the Bible in the case of Christians. More on Godel is given below in extracts from Vincent Poirier's review of the book "The continuum hypothesis" by P J Cohen, Dover 2008.

>> More comments on Godel's theorem (start) <<

[The book, The Continuum Hypothesis] proves that a long standing problem in mathematics (the Continuum Hypothesis) has no solution. What does this mean?

Most mathematicians believe in a scaled down version of Hilbert's Programme. Hilbert hoped that all of mathematics followed from a small collection of definitions and axioms, much like all of geometry was once believed to follow from Euclid's five axioms. Formal set theory, as defined by Zemerlo and Frankel, seemed to provide all the axioms needed for this task. However Kurt Gödel proved that the programme is impossible to realize: any formal system will have propositions that are possible to state but impossible to prove. In other words, no set of axioms can completely define all of mathematics.

Paul Cohen proved that the Continuum Hypothesis is one such statement.

Georg Cantor... first stated the Continuum Hypothesis and he spent years trying unsuccessfully to prove it. In the 1930s, Kurt Gödel proved that if you assumed that the Hypothesis was true, you did not contradict formal set theory.

In 1964 Paul Cohen proved that if you assumed the Hypothesis was false, you did not contradict formal set theory either. And so he shows that in the context of set theory the Continuum Hypothesis is unprovable.

You might wish to read the whole review since the above is just an extract but it is an example of Godel's theorem in action!

>> More comments on Godel's theorem (end) <<

The author mentions that nobody has ever seen God through telescopes, or met him after landing on the moon etc. (One of the early Soviet cosmonauts said the same thing.) However, no Christian with a sufficient level of education would expect to see God in those circumstances.

I do not comment on the Bayesian methods applied to determine the non-existence of God since I am not familiar with them. Others will probably comment on such use.

I enjoyed reading this book and I agree that the author has demonstrated the useful result that God cannot be infinite.