A Theology of Mathematics1
A theology of … what?!
What is Mathematics?
Where are we now?
A note on Mathematics and Post Modernity
Beauty in Mathematics
Morals in Mathematics
The Mathematical mind – strengths and weaknesses
Why do so many people dislike Mathematics?
Conclusion
Bibliography
“Numbers were beautiful things, numbers were funny things, they were without a doubt ‘God stuff’” – Mister God, This is Anna by Fynn
A theology of … what?!
The first issue raised by this subject is the surprise usually generated by putting together the two words “theology” and “mathematics”. In his opening lecture for the Christian in the Modern World course, entitled the Sacred/Secular Divide, Mark Greene asks the question ‘Who has a theology of mathematics?’ There are very few people prepared to try and answer the question; despite the fact, as he goes on to say, that we have each spent perhaps an hour a day, five days a week (during term time!) for eleven years or more in maths lessons during our schooling. Why do we have no coherent way of relating this activity to our beliefs in God? We seem to have lost the sense that God has anything to do with mathematics. Incidentally this is true of other things too – Mark is using mathematics principally as an example of the dichotomy between ‘religious’ and ‘secular’ activity that is present in contemporary Western culture.
This separation of mathematics from theology is historically odd; many of the famous names from mathematics in the past were also known for their theological writing.
Blaise Pascal (1623-62) is probably best known today either for his ‘Pensees’ which are a collection of ‘thoughts’ mostly on the subject of human suffering and faith or for ‘Pascal’s wager’ which says “If God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing[1].” However he was also an important mathematician, who laid the foundations for probability and also gave his name to the ‘Pascal triangle’ (which was actually known years before Pascal himself studied it). This is the table which starts like this:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
where each number is the sum of the two numbers in the row above it.
This triangle was the basis for Isaac Newton’s (1643–1727) work on binomial expansions – and he was another theological writer: “God created everything by number, weight and measure.[2]”. Of course he is best known for watching apples falling.
Johannes Kepler (1571-1630) is another famous astronomer, theologian and mathematician who described his work in understanding planetary orbits, which involved mathematical work on ellipses, as “thinking God’s thoughts after him[3]”.
Before him Giordano Bruno (1548-1600) was a free thinker who was eventually killed by the Inquisition – part of his crime was explorations he had done on infinite which was deemed heretical since the Catholic doctrine of the time held that only God could be infinite. Here was a darker connection between theology and mathematics. Before this, for both Arabic and classical Greek philosophers, mathematics was seen as closely coupled to religious thinking.
However for most of us today this connection is lost, so I intend to show various ways that a Christian mind can interact with the study of mathematics.
What is Mathematics?
This deceptively simple question is actually very hard to answer. I’ll try to give a brief overview without assuming too much mathematics – some of the slightly more technical bits are in separate boxes and can be skipped without problem.
The ancient world
The Egyptians knew about right-angled triangles and used this knowledge to build things like the pyramids, which are still with us today.
They knew that a piece of rope knotted into twelve equal portions could be used to measure a square corner - a bit like this:
However, as far as we know, they didn’t seem to have been able to work out why this was true; perhaps they weren’t interested in the ‘why’ since they mainly used mathematics to get things done.
Optional information: Pythagoras’s theorem.
For any right-angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides.
We know that it works because 32 + 42 = 52.
As any school child knows this is an example of ‘Pythagoras’s theorem’ which was first proved by the Greek philosopher Pythagoras (~569-475 BC). In fact the Babylonians had worked out the rule a millennium before him although they seem to have been unable to prove it.
So in this case mathematical knowledge progressed from:
-an observation that the 3-4-5 triangle has a right angle to
-a rule that a triangle with side a,b,c is right angled if a2 + b2 = c2 to
-a theorem that this rule is true for any a, b and c
Optional Information - Euclid’s postulates:
1)A straight line segment can be drawn joining any two points.
2)Any straight line segment can be extended indefinitely in a straight line.
3)Given any straight line segment, a circle can be drawn having the segment as radius and one end point as centre.
4)All right angles are congruent.
And the ‘odd man out’:
5)If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
The Greeks were fascinated by ‘pure thought’ and many of the foundations of mathematics were laid by them. One of the greatest of these Greek mathematicians was Euclid (~325-265 BC). His ‘Elements’ was one of the major attempts to provide a systematic summary of the results they discovered, and is possibly the second most translated, published and studied book ever written (after the Bible).
Euclid wanted to provide a firm foundation for mathematics and so he gave a great deal of emphasis to rigorously proving theorems from axioms. For example, in geometry his hope was that we could choose a relatively small number of key statements which every reasonable person can see are self-evidently true (these usually known as axioms or postulates) and then deduce the whole of geometry as logical consequences of these axioms.
He was able to get his starting set down to only five postulates.
He was unhappy with the fifth postulate, mostly because it was inelegant, and wanted to be able to remove it. However he was unable to prove it true from the first four and neither could he find a simpler equivalent. Over the next two millennia many mathematicians tried, and failed, to do the same.
Euclid obviously believed the lines, circles, etc. he described were those of the real world and so his mathematics was a description or codification of the nature of the universe. His views led years later to the statement “God is a geometer” which seems to have been coined by Kepler. It seems fair to say that most mathematicians – at least those from Western traditions – believed that they were discovering truth about the universe and that intuition, science and mathematics were all different views of the same thing – reality.
Shaking the foundations
Optional information – Geometries
Euclid’s fifth postulate is equivalent to the idea that there is a unique parallel to any line through a point not on the line.
There are two main flavours of non-euclidean geometry, which correspond to two different answers to the question “How many parallels are there?”. In hyperbolic geometry there is more than one parallel line and in elliptical geometry there is no such line.
Note that they all share some theorems – any theorem which you can prove using only the first four of Euclid’s axioms is true in all of these geometries.
In 1823 this world view changed forever – although most people didn’t realise it at first.
Johann Bolyai and Nikolay Lobachevsky independently discovered geometries which did not assume Euclid’s fifth postulate. This produced three main flavours of geometry – Euclidean, Hyperbolic and Elliptical.
The $64,000 question is: which one of these geometries matches the real world? The question cannot really be answered by experiment since an infinite line would take some time to construct… and the question cannot be answered from within mathematics since all three geometries are internally consistent.
Meanwhile, elsewhere in the forest…
Optional Information - A paradox of infinite
Take the natural numbers:
1, 2, 3, 4, 5, 6, …
Now double them:
2, 4, 6, 8, 10, 12, …
There is an infinite number in both rows – which row is bigger? Obviously the second row is bigger because it is double the first row.
Start again but this time throw away every other number:
2, 4, 6, 8, 10, 12, …
Now which row is bigger? Obviously the first row because we took things out to create the second row.
But wait – the resultant row in both cases is the same – how can it be both bigger and smaller than the starting row?
Things were getting confusing in the world of arithmetic too. Georg Cantor (1845-1918) was playing around with infinity.
One of the biggest problems with infinity is trying to compare it with itself – many paradoxes lie waiting to trap the unwary.
Cantor realised that there were two ways to compare the sizes of sets of things – take knives and forks for example. One way is to count the knives, count the forks and check the totals are the same. Another way is to pair up the knives and forks until you run out of pairs. If you have nothing left then you have the same number of knives and forks; if you have something left over then this tells you which you’ve got more or – knives or forks. This second method doesn’t involve remembering totals or even being able to count.
He found that using this second method of comparing sizes dealt with the paradoxes and enabled a rigorous treatment of infinity. He went on to prove that there were many different infinities with different sizes.
In particular he proved that there are more decimal numbers between 0 and 1 (i.e. ‘point-somethings’) than all the whole numbers (i.e. the numbers 1,2,3,… etc. ). He was able to demonstrate this by showing that however you pair up the decimal numbers with the integers there are always decimals left over. Nowadays almost every mathematician accepts this result, although it seems extremely unlikely when it is first explained.
He did not receive the same consequences as Giordano Bruno above, or even those of another earlier mathematician Bernhard Bolzana (a Czech theologian/mathematician who lost his teaching post at Prague in 1819 following his investigations into infinity) but he did have much opposition to his ideas from other mathematicians and philosophers.
It is likely that this opposition contributed to his eventual madness, although the subject matter of his thoughts probably had as much to do with it.
Cantor’s work on infinity threw up a hypothesis– the so-called ‘continuum hypothesis’. Much effort was spent trying to prove or disprove this hypothesis. Kurt Gödel (1906-78) proved that the hypothesis was consistent with the rest of arithmetic and then Paul Cohen (1934-) proved that assuming this hypothesis to be false was also consistent with the rest of arithmetic. So you could take it or leave it – both ways produced a completely consistent system.
Once again, like Euclid’s fifth postulate, how could you know which was ‘true’? I still remember the surprise I received while listening to a maths lecturer who was proving a theorem when he said, “My proof of this theorem will use the continuum hypothesis. If you don’t believe this hypothesis (and I don’t, but most mathematicians do) then the proof is still possible but a lot harder”. I had never thought of belief before in the context of mathematics.
You just can’t prove everything
Gödel’s most important result however is arguably the ‘incompleteness theorem’. The full result is fairly complex and the proof is quite hard to grasp. However the theorem states that, given the rules of arithmetic, there are statements of arithmetic which are true but cannot be proved. This shocking conclusion was the death knell for the attempt to build up a complete picture of all mathematical truth from basic axioms since even arithmetic, which seemed deceptively simple, was incomplete in this sense.
Where are we now?
Without a doubt the relationship between mathematics and reality is less obvious than it was. This has a big impact on how we think about mathematics theologically.
As far as I can tell there are three strands in the approaches being taken today.
1)Mathematics is an a priori truth about the real world: 2 + 2 does equal 4
2)Mathematics is an empirical construction from observing the world
3)Mathematics is a purely human construction – it ought to be an arts subject
These three basic approaches are in practice often combined in various ways, but I’ll treat them separately as I look at what a Christian mind can affirm and challenge in these different views.
A Priori
The first approach is highlighted when mathematicians talk about ‘discovering’ a result. It is a very common view, both inside and outside the world of mathematics, and is even reflected in law since mathematical theorems cannot be patented. This view is most like the traditional view that has been held almost universally, until recently, since the Greeks. Many mathematicians “really have the feeling of moving in an abstract landscape of numbers or figures that exists independently of their own attempts at exploring it[4]”
As Christians we can affirm the sense of reality in this approach. We would want to assert that absolute truth can exist - and does so in God. If part of this truth is mathematics then when we do mathematics we are in some sense ‘thinking God’s thoughts after him[5]’. In previous times people have used the existence of mathematics as part of “natural theology” attempts to prove the existence of God. Immanuel Kant (1724-1804) was one such philosopher and although many people today would reject the validity of this sort of proof it still seems to be hard to believe in the objective truth of mathematics without somehow putting some sort of god into your world view.
However we would want as Christians to challenge attempts to exalt mathematics as the ultimate truth - God may be a geometer but he is much more than just a geometer. Gödel’s incompleteness theorem stands as a reminder that mathematics cannot prove everything that is true.
Empirical
The second, empirical, approach (which seems to have similarity to the Egyptians’ view) is often associated with the philosopher John Stuart Mill (1806-73) who wrote that geometry “is built on hypothesis; that it owes to this alone the peculiar certainty supposed to distinguish it; and that in any science whatever, by reasoning from a set of hypotheses, we may obtain a body of conclusions as certain as those of geometry, that is, as strictly in accordance with the hypotheses, and as irresistibly compelling assent, on condition that those hypotheses are true[6]”. Those holding this view often see mathematics just as a tool, or a language, for doing other things – whether science or economics. Mathematics in this approach has no objective reality – two plus two is four by experiment and hence, presumably, could be proved false.
A Christian critique would want to affirm that the reason that this empirical approach works at all is because God created and sustains both the universe and ourselves. Hence it is not surprising that we can, at least in part, understand the universe. The book of Job, for example, “shows that Man was intended to argue with God[7]” – the force of the conclusion to the book is that it is a reply to Job’s questions: Job 381 “Then the Lord answered Job out of the storm”. It has often been noticed that contemporary science arose out of a world view strongly influenced by Christian belief in the order of the universe and the God-given rationality of man enabling us to comprehend it.
A Christian would also like to point out that, without a belief in God, the main problem with the empirical approach is why does mathematics work like this – Einstein said something like “the most incomprehensible thing about the world is that it is comprehensible[8]”.
Artistic
The third approach at its most extreme argues that mathematics has no external reality at all. It is simply an attempt to impose pattern onto a chaotic universe. Few people go that far; rather more would be happy to echo G.H.Hardy (1877-1947) who in his book ‘A Mathematician’s Apology’ (which he wrote towards the end of his life) says “Real mathematics”, as he referred to it, “must be justified as art if it can be justified at all.[9]”
In this approach “if geometry is not God-given, then it is people created, and how do we think about human creativity?[10]”. We can affirm the incredible richness of the creative project which is mathematics. However, many Christians might be unhappy at the ‘privatisation’ of mathematics which is an effect of denying it any external reality. The near universal agreement over mathematical truth and the way even very abstract mathematics keeps popping up years later in science are more consistent, to a Christian mind at least, with the first two approaches.
The Bible and Numbers
There are a few thoughts and principles which can be drawn from the Bible about arithmetic and rationality.
Reason and understanding are seen as gifts of God – so Nebuchadnezzar temporarily loses his reason as a punishment in Daniel 4. God expects man to reason with him (Is 118) and he is the source of understanding although his thoughts are seen as higher than the thoughts of man. There is little sign of a dichotomy between ‘faith’ and ‘reason’ in the Bible – Christians are enjoined to give a ‘reason for the hope that you have’ (1Pet315).