Mathematics between spiritual realm of experience and earthly application

February 2022

Stephan Sigler is a lecturer in mathematics, geography and the foundations of Waldorf education at the teacher training seminar in Kassel as well as a middle and upper school teacher at the Kassel Free Waldorf School.

EK | Mathematics was historically closely related to astronomy. Does it also have to do with the “heavens” as a spiritual space of human experience?

Stephan Sigler | If you look at modern astronomy today, it is essentially physics that resolves itself into mathematics at the end. Planetary orbits have long been calculable in principle and more remote “phenomena” like black holes can be explained with differential equations. Starting from antiquity, there are of course completely different lines of thought, which are based much more on the phenomena of the starry sky and its effect on people. Interestingly, this can also be experienced in pupils who, for example in class 10, perhaps for the first time in their lives, consciously look at the starry sky during their surveying practical. They lie down on their backs and become still. And indeed: if we allow ourselves to engage in such observation, it is almost inevitable to feel something like awe at the majesty of the sight.

EK | Isn’t that just a subjective feeling?

StS | Yes, of course it is subjective, but only insofar as it is felt by a subject. But the historical fact is that throughout all cultures, the phenomena of the heavens have always been seen as the direct expression of a mathematical order of divine origin in which the human being could locate themselves. The cosmos and with it, of course, the human being, were experienced as being interwoven in wisdom, if you will, with mathematics, which found its aesthetic expression, for example, in the idea of the harmony of the spheres. The harmonious order of the cosmos finds its direct expression in music already in antiquity with Pythagoras; then of course also with Kepler or in Gustav Mahler’s eighth symphony, of which he wrote that in it the universe, the circling planets and suns, begin to resound and sound. The starry sky is always in the background as an image and metaphor for the lawfulness of the world, which today has become the sober laws of nature.

EK | And the pupils experience this lawfulness of the world in mathematics?

StS | First of all, they experience within the framework of mathematics that there are indisputable connections in thinking. In mathematics, there is no dispute about the validity of mathematical statements. All mathematicians are convinced that an ultimate explanation can and will occur. It is all just a question of time. In mathematics, then, there is quite obviously something like a common “heaven”, a uniform space of human experience.

EK | Mathematics as revelation? In actual teaching, that would mean that the teacher would only have to explain mathematics well and the pupils would then follow these explanations. That seems to be a very old-fashioned picture of mathematics education.

StS | The crucial thing is that the pupils first have to do mathematics before it can become an object of reflection. This means that all mathematical phenomena are first produced through their own activity, through forming geometrical ideas, which is guided by so-called “imagination exercises”, or by performing calculations that are arranged accordingly. The “material of experience” is the consummation of our own inner movement which in real terms produces orders, patterns and laws which can then be discovered, systematised and generally formulated in symbolic language. But first it needs this initial contact with the production of mathematics in which beauty and harmony show themselves. This method can be compared very nicely with physics lessons: first, series of phenomena are revealed through experiments which are then investigated the next day with regard to the laws connecting them. This is actually an exploratory procedure, but it is guided by the staging of the experiments or the way the ideas are formed or the calculations in such a way that the essentials can express themselves in the phenomena.

EK | Rudolf Steiner also called mathematics the “preparatory school of knowledge of the spirit”. Can you explain this? Does this aspect play a role in the lessons?

StS | Rudolf Steiner showed a path that can lead to knowledge of processes and things that cannot be grasped by the senses through inner, meditative work. Now mathematics describes an object area that is just the same: not comprehensible through sensory perception! Mathematics is always only thought, never a sensory thing. We think we see a circle when one is drawn on the blackboard, but what we see is white chalk dust. But if we think of the “circularity” of the circle, we can arrive at the following formulation for example: “The geometrical location of all points that are equidistant from a point in a plane is a circle”. There are, of course, many other ways to formulate the “circularity” of the circle: all formulations point to a characteristic thought, the “concept” of the circle. It is not about the term but about this “concept”, i.e. about the “nature”, the “circularity” of the circle.

The concept can only be produced by thinking. An individual act of cognition must be carried out which takes something into regard that cannot be grasped with the senses and – as explained above – is universally valid. All mathematical objects are of this nature. Even simple numbers like a three are mental creations of the human being: the three does not lie in the three chairs that I perceive in a given situation – they are simply chairs – but in myself, who mentally combines the three chairs into a unity. This becomes even clearer with the introduction of negative numbers in class 7. Here there is still a bridge to everyday life for addition and subtraction when we think of debts, but it already breaks down for multiplication: what does it mean in terms of content if you calculate (-2) times (-3)? The matter becomes even more obvious when it comes to questions of infinity, which are the main topics in projective geometry and calculus in classes 11 and 12. Infinity is characterised precisely by the fact that it cannot be realised in space and time.

EK | Then projective geometry would be important in this respect, which is a hallmark of the Waldorf school. What is characteristic of this main lesson in class 11?

StS | That it is a hallmark is more a wish than reality. Many teachers in the upper school of Waldorf schools have not acquired an additional Waldorf educational qualification, although there are practicable in-service provisions. In this respect, this subject area is foreign to many and is simply not taught in view of the perceived and real pressure of central final examinations. This is extremely regrettable because this main lesson is one of the real highlights, especially for pupils whom we might described as less mathematical.

I would like to pick out just one educational aspect: in projective geometry, forms of thinking can be practised that go far beyond developing mathematical skills in the narrower sense. As human beings, we are psychologically and physically constituted in such a way that we primarily think of reality as being made up of the smallest particles. That is completely natural. A straight line is made up of points, the number 12 consists of twelve units and the whole world is made up of tiniest particles. Crystals of table salt, for example, are formed from sodium chloride molecules joined together. The geometrical structure of these molecules is held responsible for the fact that table salt exists as a crystal in cubic form (cf. Figures 1 and 2). Crystallography deals with the classification of all crystal forms that exist. Interestingly, that also includes all forms that are mathematically possible.

Now, in projective geometry, the world of points in space can be supplemented through the duality principle with the world of planes. What is valid in one world is also valid in the other in a dual way – in everyday language we could perhaps also say in a complementary way. Thus if we build up a salt crystal of atoms in the world of points that we are familiar with, we can also enclose the crystal in planes in a dual manner. Growth in the point world from the centre outwards corresponds in the plane world to formation from the outside inwards from the infinite periphery towards the crystal surfaces. The points of the crystal have their absolute interior in the centre, whereas the planes of the crystal have their absolute interior in the infinite periphery, which in projective geometry must be thought of as an infinitely distant plane (cf. Figures 3 and 4).

Duality guarantees that the world of points and of planes has the same structure. Whether you formulate in points or in planes – you get the same structure. The whole of crystallography can also be formulated in the world of planes.1 From a mathematical point of view, there is no reason at all to start from points – i.e. atoms. A formulation based on planes would be just as justified. We might ask which area of reality we touch on with this form of thinking in planes. Mathematics thus becomes a training ground for forms of thinking. But that the absolute inner corresponds with the infinite periphery is, of course, a strong image for the human being as being integrated into the cosmos.

EK | Such intellectual dimensions of mathematics have today rather given way to their practical application, e.g. in statistics or computer technology, even in medical research. The whole world talks about algorithms and probabilities today. Do mathematics lessons at Waldorf schools deal with these practical applications of mathematics – possibly critically?

StS | Steiner’s very early demand was that all lessons must be life lessons. The flagship in this respect is the surveying practical in class 10: well thought out and stringently planned, the abundance of spatial phenomena of a section of landscape can be confined into a map through measured and calculated access. In the process, the one-sidedness of such access can be strongly experienced through the sight lines, equipment and surveying rods which contrast with a beautiful landscape. This experience is important.

In other fields, such ideas for a life-related studies approach are unfortunately lacking – much still needs to be developed here. What is published today by universities in terms of teaching methodology in the new generations of textbooks is, in my opinion, not convincing since there is an unreflected call to formulate the whole world in mathematical models and to apply mathematical operations in order to then transfer the solution obtained back into the world. The context of meaning and the possible moral implications of such a mathematical solution cannot be appraised, if only because at heart it is always assumed that a solution represents something essential – after all, that is why it was calculated! Since the solution was obtained through mathematics, it cannot really be doubted any longer and therefore enjoys an almost unassailable status. I consider this to be a socially dangerous development, because it incapacitates people’s ability to judge. Other areas of reality are completely blanked out.

The interview was conducted by Mathias Maurer

1. Cf. Ziegler, Renatus (1998): Morphologie von Kristallformen und symmetrischen Polyedern. Kristall- und Polyedergeometrie im Lichte von Symmetrielehre und Projektiver Geometrie. Dornach: Verlag am Goetheanum (Mathematisch-astronomische Blätter. Neue Folge, Bd. 21)

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