A Theology of Mathematics
A theology of … what?! ........................................................................................ 2
What is Mathematics? ............................................................................................ 3
Where are we now? ................................................................................................ 5
A note on Mathematics and Post Modernity ......................................................... 8
Beauty in Mathematics ........................................................................................ 11
Morals in Mathematics ........................................................................................ 12
The Mathematical mind – strengths and weaknesses .......................................... 13
Why do so many people dislike Mathematics? ................................................... 14
Conclusion ........................................................................................................... 17
Bibliography ........................................................................................................ 18
“Numbers were beautiful things, numbers were funny things, they were
without a doubt ‘God stuff’” – Mister God, This is Anna by Fynn
Roger M. Orr
November 2003
CMW project
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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 believing1.” 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
1
1
2
3
4
1
3
6
1
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 him3”.
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.
“Blaise Pascal” at www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html
Bar-Ilan University Physics Dept at http://www.ph.biu.ac.il/SRP.php
3
Institute for Creation Research at http://www.icr.org/pubs/btg-a/btg-034a.htm
1
2
A Theology of Mathematics
3
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.
We know that it works because 32 + 42 = 52.
Optional information: Pythagoras’s theorem.
As any school child knows this is an example of
‘Pythagoras’s theorem’ which was first proved by For any right-angled triangle the square on the hypotenuse
the Greek philosopher Pythagoras (~569-475
equals the sum of the squares on the other two sides.
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
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
Optional Information - Euclid’s postulates:
discovered, and is possibly the second
most translated, published and studied
1) A straight line segment can be drawn joining any two points.
book ever written (after the Bible).
2) Any straight line segment can be extended indefinitely in a
Euclid wanted to provide a firm
straight line.
foundation for mathematics and so he
3) Given any straight line segment, a circle can be drawn having
the segment as radius and one end point as centre.
gave a great deal of emphasis to
4)
All right angles are congruent.
rigorously proving theorems from
axioms. For example, in geometry his
And the ‘odd man out’:
hope was that we could choose a
relatively small number of key
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
statements which every reasonable
right angles, then the two lines inevitably must intersect each
other on that side if extended far enough.
A Theology of Mathematics
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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
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.
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.
Meanwhile, elsewhere in the forest…
Things were getting confusing in the world of arithmetic too. Georg Cantor (1845-1918)
was playing around with infinity.
Optional Information - A paradox of infinite
One of the biggest problems with
infinity is trying to compare it with
Take the natural numbers:
1, 2, 3, 4, 5, 6, …
itself – many paradoxes lie waiting to
Now
double
them:
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
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?
A Theology of Mathematics
5
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. ‘pointsomethings’) 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 (190678) 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
A Theology of Mathematics
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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 it4”
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 him5’. 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 true6”. 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 God7” – the force of the
conclusion to the book is that it is a reply to Job’s questions: Job 381 “Then the Lord
4
Dehaene, p242
See note 3
6
“Logic”, II. V. I, quoted at http://www.utm.edu/research/iep/m/milljs.htm
7
‘He Came Down from Heaven, Charles Williams
5
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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
comprehensible8”.
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).
However understanding in the Bible does not arise in a vacuum – Proverbs in particular
stresses that man’s understanding and wisdom comes out of a relationship with God. “I
believe therefore I am” is probably closer to this than the Western “I think therefore I
am” of Descartes.
God creates order out of formlessness in Genesis 1; which is reflected in the structured
way it is written. For example there are two triples of days ending with the special 7th
day. God is seen as creating and sustaining the order of creation – sunrise and sunset,
summer and winter (Gen 822). The Bible sees the order and rhythm of the world as a
sign of God’s hand in it and I think this would extend to arithmetic also.
Then too the Hebrews were very interested in numbers and their symbolism. Most of
the smaller numbers were associated with specific ideas, and so their use of ‘round
numbers’ as approximations was more complicated than that of Western writers. One of
“Collected Quotes from Albert Einstein”: http://rescomp.stanford.edu/~cheshire/EinsteinQuotes.html
from ‘Understanding Analysis’ at http://community.middlebury.edu/~abbott/UA/UA1point1.html
10
http://www.stthomas.edu/cathstudies/1995/mclean.htm
8
9
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the most important numbers was seven, which spoke of completion and perfection - as
for example “the sevenfold spirit of God” (Rev 31).
The Bible sees being able to number things as giving some sort of power over them – the
importance is shown by the way all sorts of things get counted from people to drinking
vessels. David gets into trouble when he counts the Israelites in 1 Chronicles 21 which
seems to be because he is abrogating God’s right alone to know this. In the New
Testament we hear from Jesus that the ‘hairs of your head are all numbered’ (Matt 1030).
Things that could not be counted – such as the stars, the clouds or the grains of sand –
are used to express the restricted nature of man’s mind and contrast it with God’s
omniscience. When God is promising Abraham lots of descendants he compares them
to the stars, which Abraham cannot count. However in Psalm 147 God knows the
number of the stars and his understanding has no limit (literally ‘no number’). So too in
Job 117 Zophar asks Job the rhetorical question “Can you probe the limits of the
Almighty?”
The interest in numbers can however become a wish to find hidden significance in them.
This is usually known as numerology and can be found in the Cabbala of later Jewish
thought – hidden numbers in Biblical (and other) texts which reveal to the initiate the
secret code unknown to others. A similar view seems to be behind some of the extreme
interpretations of the book of Revelation and the famous number of the beast ‘666’.
Although it is undoubtedly possible that God could embed prophetic numbers in the
Biblical text it is unclear why he would have done so – the codes and numbers which
people find are generally only understandable when looking back at events once they are
completed and the events so described seem arbitrary. It can be all too easy to find
apparent connections between historical events and numbers that are in fact simply a
coincidence – there are so many events and so many ways to combine letters and
numbers that the range of possibilities becomes immense. A simple example (using the
King James version) is the ‘proof’ that Shakespeare wrote the Bible - “Psalms is the most
poetic book of the Bible. Psalm 46. Take the 46th. word from the beginning. Add the
47th. word from the end. You have revealed the true author of the Bible.11”
Additionally the bible itself does not claim to be a book of numerology and Jesus made
no references to this view of the Hebrew Old Testament despite making many
statements about it and its role.
A note on Mathematics and Post Modernity
Mathematics does not sit very well with post-modernity. There are a number of reasons
for this, the most important being the complete inability of mathematics to survive any
sort of inconsistency.
Alan Turing (best known for his code breaking in World War II) apparently attended a
series of lectures by the philosopher Wittgenstein who argued that contradictions should
not be rejected – Turing attempted to counter this view and eventually gave up attending
the lectures12. The inclusion of just one contradiction in a mathematical system allows
you to prove any result you want.
11
12
From http://www.geocities.com/Athens/Troy/4081/Shakespeare.html
Barrow2000 p300
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However, the postmodern world view has given mathematicians increased flexible about
which axioms are taken as the starting point, even though they still use the same logic to
produce results from them.
Perhaps Christian thinkers can learn something from mathematics about making a clear
separation between the choice of initial axioms from the conclusions such starting points
lead to. For example we might recognise that although in many cases starting
assumptions cannot be proved the way in which these are used to reach to a conclusion
can be challenged. As Tom Wright wrote “The church must recover…its faith in Godgiven reason, not as an independent source of authority but as the tool for thinking
clearly13”.
Logic and reason is very much a product of the conscious mind – the unconscious mind
which is so much a part of psychology does not seem to follow the same rules of logic.
One of the problems of an over-rational approach to life is that it discounts many of the
things which are more associated with the unconscious – such as symbols, myths and
poetry. These are often stressed in a postmodern approach but sometimes it seems that
the pendulum swings too far in the direction of abandoning reason. In the same way
that psychology looks to restore a healthy balance between the conscious and the
unconscious there is a proper balance between rationality and non-rationality.
One specific area where I believe there is a connection between post modernity and
mathematics is in the relatively recent exploration of so-called ‘Chaos theory’ (which is
perhaps better described as “order without regularity”) as seen in things like weather
patterns and the way taps drip. The mathematics at the foundations of this theory is not
hard but the Enlightenment mindset didn’t find it. This seems to be because people were
unable to accept that complicated behaviour could arise from simple causes and so
looked for increasingly complicated causes – the older world view understood ‘order’ in a
relatively deterministic and predictable way which did not fit with the unpredictable
nature of chaos theory.
For example, consider the very simple formula f(x) = 2x2 – 1. If you apply this
formula to its own output again and again you might expect to get some sort of pattern.
However, if you try it with a pocket calculator or a computer you’ll probably be surprised
– it looks almost random. Here for example are the last 50 numbers generated when I
started with an initial value for x of 0.78 and repeated the calculation 1000 times on a
computer.
1.5
1
0.5
0
1
-0.5
-1
-1.5
13
“That special relationship”, The Guardian, 18 Oct 2003
51
A Theology of Mathematics
It might look random – but it isn’t!
10
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Beauty in Mathematics
Beauty also has an important place in mathematics – mathematicians and theoretical
physicists have an instinctive preference for elegant theorems; Paul Dirac is one of the
best known of many who apparently preferred a beautiful theory to one which simply
complied with observations. This fits well with our understanding of the nature of the
universe from Genesis that it was initially created ‘very good’.
Here are a couple of examples of beauty in mathematics.
1) The Mandelbrot set.
This set was highly popular during the 1990s and extracts from it appeared everywhere.
This is an image of the entire set.
The set arises out of chaos theory and is generated by using a deceptively simple formula
f(z) = z2 + c which is applied again and again to its own output for different
starting values of ‘c’. The detail that is revealed when magnifying particular areas of the
set is astonishing and seems out of all proportion to the simplicity of how it is defined.
2) An incredible relationship.
Five fundamental mathematical quantities:
e (the base of natural logarithms - it also appears in probability)
 (the ratio of the circumference of a circle to its diameter)
i (the square root of minus one, much used in electrical theory)
one
zero
are related in this equation:
ei + 1 = 0
I personally find this result (and some other unlikely relationships) almost unbelievably
amazing since there seems no good reason why quantities which were independently
discovered should have such an intimate relationship.
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For many people the beauty of mathematics is a sign of the beauty which God placed in
the universe. I find it hard to see why ‘beauty’ should have any relation to ‘truth’ if the
universe were in fact meaningless.
Morals in Mathematics
The work of pure mathematics seems to have little connection with morality, other than
the basic integrity of verifying, to the best of your ability, that you have actually produced
a complete proof; which sometimes requires much more work than a partial one.
One obvious area where morals matter enormously in applied mathematics is statistics.
37.4% of statistics are made up on the spot14. Seriously though, statistics have a great
deal of power in today’s world and are used to justify decisions of all sorts. Here are a
couple of ways in which morality is relevant.
The first call is to honesty in researching and presenting statistics. “There are lies,
damned lies, and statistics15”, but statistics do not need to be untruthful - there has been
a substantial amount of work done on what must be done to ensure that statistics are as
truthful as possible and an ethical approach to statistics must be informed by this work.
If we are creating fresh statistics it is important that we take care, as best we can, that
what we present has been accurately researched and is presented without deceit. For
example, what is the difference between the two statements below?
(a) “I asked, and Bill reads the Bible although Fred doesn’t”
(b) “A recent survey found that 50% of people read the Bible”
The same facts are presented in both cases, but the second time they are presented in a
way that implies far too much. There are a lot of technical ways to prevent this sort of
falsehood – a good beginning is to ensure the number of people asked, and how they
were selected, is always included.
There are more subtle ways – both deliberate and accidental – that make statistics
misleading. As another example, take the phrase: “In a recent survey 8 out of 10 people
preferred brand X coffee”. There are many dishonest ways to get a figure like this. One
way is to keep asking groups of 10 people until, eventually, you will find by chance a
group of ten people where eight of them happen to prefer brand X. The other groups
that were surveyed are quietly forgotten. Morality here obviously means keeping all the
results and not just selecting the ones we happen to want.
Unfortunately all those who create statistics are subject to human sins of dishonesty, bias
and laziness and so we must take care when reading statistics and be ready to question
figures – which may require some work and courage on our part. Whenever we read a
statistic we should be prepared, as Darrell Huff says, to ‘talk back’ to it: “You can prod
the stuff with five simple questions, and by finding the answers avoid learning a
remarkable lot that isn’t so.16”. Sometimes the bias may be obvious, or the figure given
with so little support that it means almost nothing, but often we have to do some work
to try and check whether what is claimed is true.
You don’t really expect a footnote for this one, do you?
Mark Twain and Benjamin Disraeli are both credited with saying this
16
How to Lie With Statistics p110
14
15
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It is all too easy when presented with statistics to selectively remember and re-use the
ones that confirm our own beliefs and to pass over the ones that challenge us. Integrity
in this area means facing up to statistics that we don’t like rather than ignoring them;
without forgetting that they could be wrong. It is also tempting to find a correlation
between two figures and then to jump too quickly to deciding one is the cause and the
other is the effect. In practice it can be very hard to identify what causes what.
When using statistics from other sources it should be part of our integrity to use them
fairly, to include any estimates of accuracy, and to provide references to the original
source of the figures. Somehow people seem to be keener to do this when quoting an
author, or using verses from the Bible, than when quoting statistics.
The Mathematical mind – strengths and weaknesses
“The marble index of a mind for ever
Voyaging through strange seas of thought, alone” - William Wordsworth
Wordsworth’s picture could be describing the archetypal mathematician, someone
lacking in the normal human emotions and working principally alone. Although this is
only part of the picture I think it does show some of the potential dangers in an
excessively mathematical view of things. Mathematics is a very abstract activity, and
abstraction has been defined as ‘selective forgetting’. Five apples and five stones do not
have much in common, apart from their “fiveness”, and to do mathematics with them is
to deliberately forget almost everything about the objects themselves.
Some mathematicians become almost compulsively rational, trying to use numbers to
solve problems of human relationships and daily living. In the book “A Beautiful Mind”
(which inspired the eponymous film) we read the true story of a John Nash, a
mathematician who lost his mind to schizophrenia – and then against the odds recovered
it. One of his most important results, and the one for which he received the Nobel prize,
was in game theory which is an attempt to provide a systematic theory of rational human
behaviour by focusing on the playing of games. However, after his recovery, he comes
to a life in which “thought and emotion are more closely entwined … relationships more
symmetrical … he has become a great deal more than he ever was17”.
Rationality believes that there are reasons for the world and one of the strengths of a
mathematical mind is the quest for finding meaning in the world, even where it is not
intuitive. However this can decay to become the view that everything has a secret meaning.
(This also seems to be related to the numerological use of the Bible mentioned earlier.)
When John Nash suffered from schizophrenia he had to wrestle against beliefs that
Russians, or aliens, were communicating with him through patterns of numbers in
newspapers, etc. that others could not find. The temptation to believe that there are
hidden meanings known only to the initiated – of which we are of course a part - is a
hard one to resist for many people.
The fifth personality type in the Enneagram model of personality, the ‘Observer’, has a
natural attraction to systems such as mathematics that “explain universal principles of
interaction18” since this type of person tends to find the outside world and emotions
17
18
A Beautiful Mind p388
“The Enneagram”, Helen Palmer, p206
A Theology of Mathematics
14
threatening and systems of thought can provide control over this fear. The ‘INTP’
temperament in the Myers-Briggs scheme describes a similar type of person.
The strengths of this approach include being able to remain objective even when
personally involved in a situation and being able to deduce principles from experiences.
The weaknesses are typically some reluctance to engage in relationships and shyness
about contributing thoughts unless the person is certain they know ‘almost everything’
about the subject.
A Christian viewpoint of the weaknesses of this personality type would want to stress
that the Bible reveals both a God in relationship – perhaps most clearly in Jesus’ prayer
in John 17 - and also a God prepared to engage in this world’s troubles even to the
extent of becoming incarnate as a human being. So the mathematical mind must be
reminded that relationships and vulnerability are very important characteristics and to be
developed as part of each person’s call to become fully human. G.K. Chesterton’s adage
“If a thing’s worth doing it’s worth doing badly19” can help provide courage to contribute
even when the person feels inadequate.
Hermann Hesse’s great novel “The Glass Bead Game” is a modern parable of this – it
describes a group of people who seek to explain everything through mathematical
symbols and repudiate emotion and strong relationships. The hero of the story, Joseph
Knecht, defects from this order, which he leads, and eventually gives up his life in
sacrifice for a fellow human being.
Of course, fortunately, not all mathematicians have this personality but it does seem that
some of the same strengths and weaknesses will apply to many of them. Whether it is an
explanation for taking up mathematics or an effect of studying it is hard to tell.
Why do so many people dislike Mathematics?
The converse of a mathematical mind is innumeracy: “a failure to deal comfortably with
the fundamental notion of number and chance20”. This is quite common in our society,
and people even take a perverse pride in being bad at maths. This is both strange, since
few people take pride in illiteracy or bad spelling, and unfortunate, since so much of our
society relies on familiarity with numbers.
W.W.Sawyer wrote “The two main conditions of success in any sort of work are interest
and confidence21”. Sadly in many cases people lack not just one but both of these
conditions when it comes to mathematics. This is most obvious in school where maths
is often a problematic subject but in many cases this lack is still present, albeit hidden,
once formal lessons are over since contact with mathematics can simply be avoided.
Many people, highly competent in many areas, fear mathematics.
A Christian would want to begin with addressing this fear. “Perfect love casts out fear”
(John 418) applies to many things, and this should also include mathematics. A love of
God and of his creation can stimulate interest in history, geography, science and so on –
including mathematics. It can certainly give us a theological backing for looking at it.
19
The Columbia World of Quotations, http://www.bartleby.com/66/49/12249.html
‘Innumeracy’, John Allen Paulos, p ix
21
“Mathematician’s Delight”, p40
20
A Theology of Mathematics
15
Reason, which perhaps reaches its purest form in mathematics, is a gift of God and we
can celebrate it. “Mr God, This is Anna” is a wonderful book which tells a story about
Anna, a small girl with insatiable curiosity and able to see things from unexpected points
of view. Maths for her was fun, a puzzle, a branch of theology and a game for sharing.
Her enthusiasm and playfulness is exciting but she is unusual – few children would be
able to cope with so many abstract concepts so young.
The use of abstract ideas is one of the places where lack of interest and confidence arises.
One of the problems with mathematics is that it does become increasingly abstract
throughout schooling. A classical model of child development designates the pre-teen
period as a key place for the rise of abstract thought. However children’s development is
very individual and so some will be earlier and some later in learning to deal with
abstraction. On top of this different people have more or less love of abstraction even
when capable of it. So it can be helpful to give concrete examples in mathematics to help
people retain interest; some teaching is better than others at providing this. In addition, a
wide range of such examples is helpful since each may demonstrate different facets of the
subject. Someone who has only seen examples of counting with beads or blocks of
wood may find negative numbers really hard to understand – has anybody seen “–3”
beads? The concept of negative numbers should be a lot easier to understand for
someone who has played with another example, such as distance, where negative
numbers have a more obvious use – if “4” means four miles north then “–4” has an
obvious interpretation as four miles south.
Picking good examples can help make it fun as well – but it is important to avoid
contrived examples. The traditional problem of how long it will take someone to fill a
bath with both taps running and the plug out was dealt with swiftly in Anna’s world by
“some mothers do have ‘em – and they live22”. This is an example of the rigidity that
sometimes dogs the teaching of mathematics – the child is expected to solve the maths
problem by using often unspoken rules. These rules are needed to find the right problem
in mathematics within the story – there may be more than one way of ‘solving’ the story
but only one will be marked as correct. The rules may also apply inside mathematics
itself – for example a poor teacher might insist on their preferred way of doing long
division even when the child already knows another way that also works. Even
something as simple as the “times tables” has more than one way of doing it – the
Chinese teach children how to swap the numbers round when multiplying so the smaller
number always comes first. This means they don’t need to learn the cases where the first
number is bigger than the second, so their tables are in total just over half the size of the
Western ones.
Rigidity like that above can lead to children doing maths ‘by rote’ and simply
manipulating symbols without really thinking about what they mean. As an example of
this sort of error consider:
1/2 + 1/3 = 2/5
where the person doing the sum has added the ‘tops’ and ‘bottoms’ of the fractions
together by misapplying the rule for multiplying fractions to adding them.
A wrong answer like this would not be accepted by someone with a mental image of
adding together half a pie and a third of a pie.
22
“Mr God This is Anna” p123
A Theology of Mathematics
16
Maths can be seen as a tool, a tool which is useful for solving things. A small number of
people have the reverse of this– everything is mathematics and so it is the tool. This
approach can seem effective as in the proverb “To a man with a hammer every problem
is a nail”. For someone like this it is more important to stress other aspects of life, things
which fall outside the area of pure reason such as love and intuition.
Unlike some other subjects mathematics is continually building on what is already
known. This means that if something causes problems the underlying reason might be
an unlearnt lesson from earlier on – revisiting this subject can both help to regain
confidence (since the problems here can be solved) and then mean the newer subject
matter suddenly seems to ‘click’.
Man’s creativity comes out all over mathematics if it is allowed to. Children count on
their fingers and often come up with ways to add and subtract small numbers on their
fingers. Later on pictures and graphs can help make numbers come to life. The
“number line”
...___________________________________________...
|
|
|
|
|
|
|
|
|
|
|
-5 -4 -3 -2 -1
0
1
2
3
4
5
is a good way to see some of the secrets of small numbers, and adding and subtracting
them. A calculator can be a bit like a map to explore the bigger numbers – it can provide
games and puzzles with numbers. These do not
A simple calculator game
necessarily need explaining – like Jesus telling a
good parable the ‘itch’ of the unsolved problem
11 – 3 x 3 = 2
can get under the skin and lead the person to
ask the right questions for themselves.
1111 – 33 x 33 = 22
111111 – 333 x 333 = 222
Numbers can be seen as fun, as known
inhabitants of the mental world. Some people
make a few friends, some make a lot; it seems the same with numbers. There is a wellknown story of the Indian mathematician Ramanujan which shows his enormous love of
numbers. He was in hospital in Oxford, and the English mathematician Hardy was
visiting him but found it hard to find enough topics of conversation. On arrival one day
Hardy started the conversation by remarking on the boring number of the cab that he
had come in – 1729. Ramanujan instantly replied that it was not a boring number – it
was the smallest number which can be formed by adding two cubes in two different
ways. (The two ways are 13 + 123 and 93 + 103).
Ignorance in mathematics is not treated in the same way as in other areas “Few educated
people admit to being completely unacquainted with the names Shakespeare, Dante,
Goethe, yet most willingly confess their ignorance of Gauss, Euler or Laplace23”. This is
not a good state of affairs and I think that there are many things which can be done,
some more explicitly Christian than others, to encourage people of all ages but perhaps
particularly during school years that mathematics need not be boring but can even be
enthralling.
23
Innumeracy p77
A Theology of Mathematics
17
Conclusion
I have tried to show that there are various links between theology and mathematics but
that, like many such links, the privatisation of religion in the West has greatly reduced the
interest in and perception of these links.
There are philosophical links between belief in God and the existence and place of
mathematics both ‘pure’ and ‘applied’, and I have tried to provide some theological views
on three main theories of what mathematics is. I myself do not believe that any single
one of the three theories is a complete explanation of mathematics – it seems to me to
contain elements of a priori truth, empirical discovery and creativity. However I do
believe that there are important things to be said from a theological viewpoint about
every one of these approaches.
There are some moral links, such as those in statistics where temptations to laziness or
bias can lead people into error.
Finally I hope I have shown how some theological basis to the subject can help people to
lose their fear of mathematics and hopefully even come to enjoy it.
A Theology of Mathematics
18
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