This article discussed the question ‘Does God speak through the language of mathematics?’ For centuries, mathematicians with different religious backgrounds would have answered this question in the affirmative. Due to changes in mathematics from the 19th century onwards, this question cannot be answered as easily as it used to be. If one regards mathematical concepts as creations of the human mind, it is difficult to argue that mathematical formulae exist in a divine mind. The article argued that there were traces of the divine in mathematics. Six kinds of traces were explained: (1) the existence of indisputable truth, (2) the existence of beauty, (3) the importance of community, (4) rational speaking about infinity, (5) the discovery that speaking about unseen and abstract objects is reasonable and (6) the unreasonable effectiveness of mathematics. In practice, traces (1), (2) and (6) are probably the most convincing.

This article is very much interdisciplinary as it combines mathematics and theology, especially the philosophy of mathematics and systematic theology.

This article is about how traces of the divine can be found in the language of mathematics. The title alludes to Psalm 19:1: ‘The heavens declare the glory of God’, which is part of the Jewish and Christian Bible.

Obviously, mathematics is some kind of language (Livio

The question to be investigated in this article is as follows:

In this article, I will not treat mathematical theistic proofs such as Gödel^{1}^{2}

This article develops some ideas that I presented in two previous articles (Kessler ^{3}

In the past, strong links between mathematics and the divine have always been assumed; see, for example, the historical study Koetsier and Bergmann (eds. ^{4}

In the Classic Greek tradition, theology and mathematics were close to each other (Phillips

To give an example of the close link between mathematics and theology some centuries ago, we look at the beliefs of some important mathematicians of the 17th century, the start of the scientific age and of modern mathematics. In the Europe of the 17th century, it was quite common to believe in the Christian God, and this is also true for the most brilliant European mathematicians of the 17th century. It is, of course, debatable that mathematician was most important at a given time; for example, Stewart (

Descartes provided two theistic proofs in his

In a world where mathematical truths were seen as part of the divine mind, it was easy to argue that God reveals himself through mathematics. The quotation from Edward Everett (1794–1865) gives a poetic description of this viewpoint:

In the pure mathematics, we contemplate absolute truths which existed in the divine mind before the morning stars sang together, and which will continue to exist there when the last of their radiant host shall have fallen from heaven. (quoted in Hersh

Similar statements are known from mathematicians with a different religious background. For example, the remarkable Indian mathematician S. Ramanujan (1887–1920) once noted: ‘An equation means nothing to me unless it expresses a thought of God’ (quoted in Kanigel

As the above examples show, the validity of Euclidean geometry used to be regarded as indisputable and was often taken as an analogy for certainty in theological issues. ‘Geometry served from the time of Plato as proof that certainty is possible in human knowledge – including religious certainty’ (Hersh

The angle-sum theorem teaches that the angles in a triangle add up to 180°. This theorem was often used as an example of an absolutely certain statement. For example, Descartes (

But this theorem does not hold in non-Euclidean geometry. The sum of the angles in a triangle can be more than 180° (elliptic or Riemannian geometry) or less than 180° (hyperbolic geometry).^{5}

^{6}

The new situation was that the mathematicians would offer different sorts of geometry from which the physicists had to choose the geometry, which would be most appropriate for studying the physical world. This new situation challenged the understanding of mathematics as a divine science.

The fact that one could select a different set of axioms and construct a different type of geometry raised for the first time the suspicion that mathematics is, after all, a human invention, rather than a discovery of truths that exist independently of the human mind. (Livio

At this point, it became obvious that intuition alone is not sufficient for mathematical certainty. Cantor’s set theory was a candidate for the foundation of mathematics. But then paradoxes in the set theory were discovered by Zermelo in 1899 and by Russell in 1901 independently of each other. This finally led to three different schools of thought on how to lay a good foundation for mathematics: logicism, formalism and intuitionism (Shapiro

Actually, the question of the foundation of mathematics has no clear answer to this day (Hersh

Although Platonism is no longer considered an up-to-date epistemology or philosophy of science, it is still alive among working mathematicians. A crucial question to mathematicians is ‘is mathematics created or discovered?’ In 1940, the renowned British mathematician G.F. Hardy (1877–1947) wrote about the ‘mathematical reality’, admitting that there is no sort of agreement on the nature of this reality (Hardy

Classes and concepts may … be considered as real objects … It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence. (quoted in Shapiro

A small survey done by Pamela Aschbacher, the spouse of an American mathematician shows that this is probably the thinking of most working mathematicians. Aschbacher (

Harrison (^{7}

As rightly noted by Hersh (

The working mathematician is a Platonist on weekdays, a formalist on weekends. On weekdays, when doing mathematics, he’s a Platonist, convinced he’s dealing with an objective reality whose properties he’s trying to determine. On weekends, if challenged to give a philosophical account of this reality, it’s easiest to pretend he doesn’t believe in it. He plays formalist, and pretends mathematics is a meaningless game. (Hersh ^{8}

This attitude could be called ‘pragmatic Platonism’ (Kessler

Hersh has a different view in mathematics. He sees mathematics as part of human culture and history and regards mathematics as a ‘socio-historical reality’ (Hersh

I think that Livio presents a well-balanced answer to the question of whether mathematics is created or discovered.

A mathematician might invent a mathematical concept like a ‘prime ideal’or a ‘crossed-product order’ (Kessler

Without digging deeper into the ontology of mathematics, I just want to state that I have much sympathy for Popper’s model of the three worlds. There is the physical world (world 1), the mental or psychological world (world 2), which is our inner world, inaccessible from outside and there is a third world (world 3), consisting of ‘the products of the human mind’ (Popper

Already in 1918, the German logician Gottlob Frege (1848–1925) published a similar idea in his essay

That seems to be the result: Thoughts are neither things of the external world nor conceptions. A third realm must be recognised. (Frege ^{9}

A thought in Frege’s sense is something, which is either true or false.^{10}^{11}

Please note that the conclusions in the next section do not presuppose the existence of world 3. I hope that my arguments will also convince readers who do not share my view on the ontology of mathematics.

How can it be argued today that God speaks through the language of mathematics? Although a pragmatic Platonism is still alive among mathematicians, I will not build on it in the following section, because I think that mathematics is a combination of inventions and discoveries, see above.^{12}

I do argue that some features of mathematics hint at divine attributes. I am not saying that one can detect God within mathematics, but that one can at least find traces of the divine. Reflections by mathematicians on their work (like in Cassaza et al.

Let us start with a personal remark by the journalist Masha Gessen. In her work on the mathematician Grigori Perelman, she also shares her own experiences with math clubs in the former Soviet Union. ‘To a Soviet child, the after-school math club was a miracle’ (Gessen

Most importantly, in mathematics, one can learn that there is an indisputable truth. A mathematical formula is either true or false. Every mathematician would agree that Fermat’s Last Theorem actually was true even before Andrew Wiles proved it in 1994 (Singh

The detection of indisputable truth might lead people to look for the source of truth, which is linked to God, at least according to monotheistic religions (see, e.g. Job 28:23 in the Jewish-Christian Bible, Jn 14:6 in the New Testament).

In 1907, the British mathematician Bertrand Russell (1872–1970) wrote ‘Mathematics, rightly viewed, possesses not only truth, but supreme beauty – a beauty cold and austere’ (Russell ^{13}

When people experience beauty in mathematics, they might start wondering about the source of this beauty. The feeling might be comparable to arriving at the top of a high mountain, enjoying the beauty of the view. At such moments one is reminded of Psalm 19:1: ‘The heavens declare the glory of God’.^{14}

Mathematics shows that there is beauty and glory, and the Bible links these attributes to God (e.g. Psalm 19:1, Isaiah 33:17, cf. Von Balthasar’s seven volumes on the glory of God, Von Balthasar

Because human beings are social beings, there are many human activities demonstrating the importance of community. And each science has its own scientific community. But I think that the previous aspect of mathematical beauty (3.2) has an interesting side effect.

Mathematics is a product of a community contributing together to create beauty. Usually, a piece of art or a symphony is the product of an individual. And in the case of a cathedral where many people have contributed at different times (632 years for the Cologne Cathedral!), they at least have to work at the same location. In mathematics, an invisible community consisting of many people in different places and at different times jointly contribute to the beauty of mathematics. In the case of Fermat’s Last Theorem, it took 358 years (Singh

In his personal account, Boas (

This communal contribution to beauty has interesting counterparts in Jewish-Christian theology. The Song of Songs in the Jewish Bible has also been interpreted as a metaphor for a marriage between God and His people. This metaphor is taken up in the New Testament calling the church ‘Christ’s bride’ (Eph 5:31–32, 2 Cor 11:2). Note that this metaphor does not refer to a mystic union between God and an individual; the whole community of believers is involved. According to this Jewish-Christian metaphor, God is looking for community.

In addition, there is the Christian doctrine of the Trinity. According to this doctrine, there is already community within God; thus community could be considered a divine attribute, at least in Christian theology (see the wonderful icon The Trinity, painted in 1410 by Andrei Rublev).

There has been a long dispute about the understanding of infinity (Hilbert

This discovery shows that it is reasonable to speak about infinity, thus demonstrating that speaking about an infinite God is not senseless per se.

Studying different degrees of infinity might also show the mathematician that there is something, which is provably higher than him or herself. We only know a subsection of all infinite sets, and this is not only because our IQ might be too small, but because the structure of the system of thought has limitations.^{15}

In her essay, Victoria Harrison argues that ‘realism about mathematical objects can provide a model for thinking about realism within theology’ (Harrison

The sections above dealt with mathematics only, looking for traces of the divine in the pure world of mathematics. This section will also look at the physical world and how mathematics is used to describe it. Physics Nobel laureate Eugene Wigner (1902–1995) once said: ‘The miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve’ (quoted in Livio

But there is also a

The issue of chance was long seen as a theological problem. How can the existence of chance be combined with a rational and omniscient God? In his book on the

Einstein raised the question: ‘How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?’ (quoted in Livio ^{16}

Creation theology as understood in the Judeo-Christian tradition offers an explanatory model for the phenomenon of ‘the unreasonable effectiveness’. God created the world with wisdom (Pr 8:22–31) and He created human beings in His own image (Gn 1:26). Thus, human beings are enabled to discover the laws that God put into His creation. I am not saying that the study of mathematics will automatically lead people to believe in creation theology. But obviously, one can find traces in mathematics that might point to the Creator-God.

The question of this article was ‘Does God speak through the language of mathematics?’ Being a mathematician and a Christian theologian, I am convinced that the answer should be ‘yes’. But I am also aware that the evidence for this answer is not as obvious as it was several centuries ago. The changes in mathematics made it necessary to rethink the evidence for this answer. We have learned that mathematical objects are created by human beings. Thus, it is not possible to argue that mathematical formulae exist in a divine mind. However, I think that there are hints in mathematics that seem to point towards attributes of God.

I have mentioned six traces of the divine: (1) the existence of indisputable truth, (2) the existence of beauty, (3) the importance of community, (4) rational speaking about infinity, (5) the discovery that speaking about unseen and abstract objects is reasonable and (6) the unreasonable effectiveness of mathematics. I think that each of these arguments has a certain weight, but I do not think that they have the same weight. If one looks at the current discussion, arguments (1), (2) and (6) are probably the most convincing.

The author declares that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.

V.K. is the sole author of this article.

This article followed all ethical standards for research without direct contact with human or animal subjects.

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

Data sharing is not applicable to this article as no new data were created or analysed in this study.

The views and opinions expressed in this article are those of the author and do not necessarily reflect the official policy or position of any affiliated agency of the author.

Gödel noted his proof on two pages in 1970 but kept it secret until his death, see Bromand and Kreis (2013:483–491).

See Poincaré’s statement: ‘The object of mathematical theories is not to reveal to us the real nature of things’ (Poincaré

In Kessler (

Jonas’ (

See Figure 41 in Livio

This was Kant’s position presented in 1781 in his

See similar remarks on Hersh (

This illustrative description is also quoted in Livio (

Original quote:

The British mathematician Penrose also speaks about three worlds. But in his case, the world containing mathematics is closer to the Platonic realm (Penrose

Note that in Kessler (

Some of Atijah’s talks are still available on YouTube.

Actually, in the Alps, there are several summit crosses quoting Psalm 19:1.

I thank my former colleague Prof. Albrecht Beutelspacher, Giessen, for drawing my attention to this; see his e-mail 8 August 2020.

Livio (