This article is transdisciplinary in that it touches on the disciplines of both mathematics and theology. It is about the mysterious link between truth and beauty in mathematics. It uses the theological pattern laid by the practical theologian Bohren in his book,

This article is transdisciplinary, touching on the disciplines of both mathematics and theology. It is about the striking links between the true, the good and the beautiful. A consideration of these issues leads to the following question: Does mathematics also have spiritual aspects? There have been some publications about spirituality in natural science, but only a few on spirituality in mathematics.

This article is also autobiographical. I write from the perspective of mathematics, which I studied initially, as well as from the perspective of theology, which I studied in my mid-30s. This article has a strong subjective component, as do the publications of the mathematicians Hardy (

Occasionally, the Bible draws links between the good and the beautiful. Often, the faith heroes of the Old Testament are also described as beautiful men or women, for example, Sarah (Gn 12:11, 14), Joseph (Gn 29:17), the child Moses (Ex 2:2), King David (1 Sm 16:18), Queen Esther (Es 2:7) and Daniel and his friends (Dn 1:6, 15).^{1}

In mathematics, there is a striking correlation between truth and beauty; in other words, ‘beautiful = true’.^{2}

In this article, I walk in the footsteps of the practical theologian, Rudolf Bohren. Bohren (1920–2010) was of Swiss origin, but he spent most of his academic career in Germany, especially in Heidelberg. In his book, ^{3}

I argue that we could also include mathematics in this list. Walking in the footsteps of Bohren, my thesis for this article is

Mathematics has always fascinated me (and still does). My conversion to the Christian faith came much later. I opted for the study of mathematics because I was looking for eternal truths. I wanted to learn something that is true –

A similar drive led me to the study of theology. I was looking for eternal – even transcendental – truth. As a trained mathematician, I was inclined to apply the mathematical method to theology, and I regarded the Bible as an axiom system. (An axiom is a statement that is taken to be true.) Logical reasoning will then lead us to further propositions, based on this axiom system. In mathematical axiom systems, it is assumed that they do not lead to contradictions – otherwise one mathematician might prove a proposition

When I applied this mathematical approach to biblical exegesis, it had the following effect: Whenever I discovered propositions in the Bible that seemed to contradict one another, I tried to solve this contradiction. This was important for me because, as a mathematician, I cannot trust an axiom system that leads to contradictions.

Since then, I have discovered that the tensions in the Bible also have an appealing quality, bearing witness to the fact that the Bible is full of life. But this was a long journey for me. Today I can see these tensions as ‘creative tensions’, a term that was programmatic for South African missiologist David Bosch (Kritzinger & Saayman

Retrospectively, I noticed that parallel to my search for truth, I also was searching for beauty. This connection became apparent to me while we, students, discussed the validity of a mathematical formula. Our mathematics professor then said: ‘This formula cannot be true; it is just not beautiful enough’. He said this while winking, but this sentence contains a fundamental truth. In mathematics there is a remarkable striking link between beauty and truth.

People might argue that beauty is very subjective. I will not even try to define ‘beauty’. Nor will I try to define ‘truth’.^{4}

I personally regarded pure mathematics as more elegant and beautiful than applied mathematics. Hence, I was more attracted to pure mathematics, and I completed my PhD in algebraic number theory.

Quotations from other mathematicians provide some evidence that my personal search for truth and beauty is quite typical of mathematicians. The famous mathematician Michael Atiyah (

Aschbacher (

His reasons were ‘the beauty of math’ and the ‘joy of algebraic geometry’. At the end of his talk, he thanked his wife of 28 years, who had given him the peace to do his research. (p. 28)

‘She let me do my research’ is the best compliment a mathematician can offer to his wife.

This section is about the beauty

Aschbacher (

But there are some mathematicians who have a different view of this. The German mathematician Hasse calls Leibniz ‘a creator of differential analysis’, not only a discoverer.^{5}

Several mathematicians feel committed to the beauty of mathematics. Plato (428/27–348/47 BC) was one of the first to speak about the link between mathematics and beauty (Heisenberg ^{6}

The Scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it; and he takes pleasure in it because it is beautiful. (Chandrasekhar

The German mathematician and theoretical physicist Hermann Weyl (1885–1955) is quoted by Dyson as saying: ‘My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful’ (in Chandrasekhar

The famous physicist Albert Einstein (1879–1955) once wrote in the

In 1940, the English mathematician GH Hardy (1877–1947) published an essay reflecting on his life as a mathematician and making a case for aesthetics in mathematics:^{7}

The mathematician’s patterns, like the painter’s or the poet’s, must be

The Austrian mathematician Emil Artin (1898–1962) is known for treating mathematics as art (in Borwein ^{8}

The above remarks hold for ‘real mathematics’ (Hardy

A wonderful example of real mathematics is Euclid’s proof that there must be infinitely many prime numbers. This proof can actually be taught at school level. It uses the interesting concept of an ‘indirect proof’, also called ‘proof by contradiction’ or ^{9}^{10}

It should be noted that Hardy, Hasse and Artin were number theorists. Number theory is a very beautiful theory, which has been virtually useless for many centuries. Hardy loved this uselessness because number theory had ‘no effects on war’ (p. 140) (this statement is no longer valid because of the application of number theory in modern cryptography). Mathematical beauty might be less important to other mathematicians than it is to number theorists.

These days Sir Michael Atiyah (born 1929) is a popular speaker on beauty in mathematics (see Atiyah

Results show that the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain, … as the experience of beauty derived from other sources. (p. 1)

What makes a mathematical theorem beautiful? The physicist Poincaré mentions ‘simplicity and vastness’ as criteria for beauty (in Chandrasekhar ^{11}^{12}

In my opinion, simplicity is a main criterion for mathematical beauty.^{13}^{14}^{2}?’ is a rhetorical question that the French philosopher Comte-Sponville (

In his short essay, Atiyah (

Hardy (

Atiyah (

Now, let us have a look at the most beautiful mathematical formula to illustrate these criteria.

The ‘beauty competition’ held by Zeki et al. delivered a clear winner. The participants, 16 mathematicians, were given 60 mathematical formulae to rate on a scale of -5 (ugly) to +5 (beautiful) the beauty of each formula (Zeki et al.

This can be attributed to the Swiss mathematician Leonhard Euler (1707–1783), and it contains exactly five numbers: 0, 1, π,

The number 0 is the neutral element of addition (‘add 0 and nothing happens’).

The number 1 is the neutral element of multiplication (‘multiply with 1 and nothing happens’).

The number π = 3.14159… has been introduced as the ratio of a circle’s circumference to its diameter^{15}

The number

The ^{2} = −1, leading to complex numbers.

The mathematical constants π and ^{16}

Thus, it comes as a surprise that raising ^{i}^{π}

Despite the enthusiasm for the beauty within mathematics and the mysterious link between mathematical beauty and mathematical truth, it should also be noted that an ugly formula is not necessarily false. In Zeki’s study, it was found that the majority of the participating mathematicians rated a given formula as ugly, even though it was a valid one (Zeki et al.

The beauty of mathematics gives those who are responsive to it something for the heart and the soul (Hasse

The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry. (p. 60)

Russell’s emotions look a bit similar to the emotions, which were called

You must have felt this too: the almost frightening simplicity and wholeness of the relationships which nature suddenly spreads out before us and for which none of us was in the least prepared. (Chandrasekhar

The mathematician Watson spent several years with the famous Indian mathematician Ramanujan (1887–1920) and later reported that a special mathematical formula from Ramanujan ‘gives me a thrill which is indistinguishable from the thrill which I feel when I enter the Sagrestia Nuova of Capelle Medicee and see before me the austere beauty’ of Michelangelo’s art works (in Chandrasekhar

The US scholar Witz (

Two of the participants in Witz’s study, both PhD students in mathematics, even speak about ‘epiphanies’ in the context of mathematics (Witz

Other mathematicians might connect their spiritual experiences with other religions or with no religion at all. But some personal accounts from Cassaza et al. (

It would be an interesting research topic to investigate the spirituality in mathematics in greater detail in order to explore the following question: What is it that a person seems to find in mathematics, but not in any other subject?

In mathematics, we find a remarkable correlation between truth and beauty. The Bible testifies to the correlation between ethics and beauty. But in both cases, there are possible exceptions: ugly mathematical formulae that are nevertheless true and physically beautiful men and women who do bad things.

Mathematicians such as Hardy, Artin, Hasse and Atiyah regard mathematics as both art and science. This is exactly how Bohren (

Mathematicians disagree about whether mathematical theories are created or discovered. The majority of mathematicians regard mathematical reality as something outside of us. If we share this view, we could argue: Every time a mathematician discovers a beautiful formula, they discover something that God created. Thus, the mathematician contributes to the visibility of God’s beauty in his creation. This would correspond with Bohren’s viewpoint that God becomes beautiful in the creation (Bohren

If we reflect that the Spirit of God is the only fountain of truth, we will be careful, as we would avoid offering insult to him, not to reject or condemn truth wherever it appears. In despising the gift we insult the Giver. … But if the Lord has been pleased to assist us by the work and ministry of the ungodly in physics, dialectics, mathematics, and other similar sciences … (

Therefore, I trust that there is now enough evidence for my initial statement:

One might critically remark that, for some, mathematics has become a substitute for Christianity, even constituting a religion in its own right. As these mathematicians enjoy spiritual experiences while doing mathematics, they might not need to look for spirituality elsewhere. It is a well-known temptation to forget about the creator if the beauty of the creation is so fascinating. But this does not belie the fact that God becomes beautiful in mathematics.

There has been very little research on spirituality in mathematics. Nevertheless, there are some known examples where mathematicians have reported spiritual experiences while doing mathematics, and it would be interesting to follow up on these. It would also be useful to ask the following questions: What is the special element in mathematics that cannot be found in any other subject? And how can we categorise the spiritual experiences in mathematics?

After reading this article at UNISA,^{17}

I hope that this paper will not only be regarded as a contribution to truth, but that it might also contribute to beauty itself. As quoted above, one criterion of beauty in mathematics is ‘the unexpected links between apparently quite different parts of mathematics’ (Atiyah

The author declares that he or she has no financial or personal relationships which may have inappropriately influenced him or her in writing this article.

Interestingly, there are no such examples in the New Testament.

This connection is also expressed in the Latin proverb

There are not many scholars outside the German-speaking countries who quote Bohren. Heitink (

Actually, I am not aware of any convincing definition of ‘truth’. Each attempt to define ‘truth’ already seems to presuppose an understanding of truth.

Original quote:

There is some melancholy in this essay, firstly because Hardy (

Actually, Hasse used the German word

Atiyah (

Hardy (

Original quote:

He uses the German words

The well-known saying ‘

While pursuing my PhD in mathematics, the criterion of simplicity led me to the correct formula. I studied a concrete example, did the necessary computations and suddenly I ‘saw’ the general formula that would fit into this example. Then I proved this general formula (Kessler

The builders of the temple in Jerusalem worked with the approximation 3 (which was also common in Babylonia). According to 1 Kings 7:23, the sea of cast metal had a diameter of 10 cubits and a circumference of 30 cubits.

A number is described as ‘transcendent’ if there is no algebraic equation with rational numbers, so that this number would solve this algebraic equation. The transcendence of π and

14 November 2017, at the invitation of Prof. Christo Lombaard, Department of Christian Spirituality, Church History and Missiology.