Mathematics conference not only for mathematicians
The mathematics conference from 8 to 10 October 2025 follows a statement by Rudolf Steiner that mathematical thinking is the very method of anthroposophical spiritual science because it produces supersensible content.
‘What clairvoyance achieves at a higher level can be studied by anyone who engages in mathematical thinking.’ This quotation is taken from a lecture Rudolf Steiner gave in The Hague (NL) on 8 April 1922 (GA 82). In this lesser known lecture Rudolf Steiner explains why engaging in mathematics is the very method of anthroposophical spiritual science. He does not refer to mathematics as content but to doing maths as a method that produces supersensible content. The scientific nature of a discipline is sometimes judged by its ability to provide measurable, countable and weighable results. Other qualities become less significant and are seen as ‘secondary’ (John Locke).
Even the concept of a number is supersensible
What is often forgotten is that even the concept of a number is purely supersensible. One cannot see, taste, smell, hear, touch or experience them in any other way through the senses. A person with true dyscalculia if you show him five beans will only ever see beans but never the number five. If you look at them closely, even simple numbers are supersensible. If you don’t have dyscalculia and are able to think a number, you can apply this supersensible concept to the sensory world and count beans, for instance. But pure mathematics exists in a supersensible world where clairvoyance is at home. ‘If we approach mathematics with the right attitude, we can see in the mathematician’s attitude when mathematizing the pattern for what is required for higher, supersensible intuition, because mathematics is simply the first stage on the way to such intuition’ (GA 82, 8 April 1922).
In his lecture in The Hague, Rudolf Steiner explained in detail how we generate the three spatial dimensions first out of ourselves and then find them again in the outside world by shaping them in a movement that proceeds from the inside out. ‘What we see as mathematical structures of space is supersensible perception’ (ibid.).
He then recommended to extend the mathematical method to other fields for ‘just as we form the three special dimensions out of ourselves, we also place the sensations rising up in us outside of ourselves.’ We achieve this by getting ‘to know the sensory world which we usually know as a world of effects as a world formed by ourselves.’ We would be surrounded by a world of intermingling colours and tones. We would speak of an objectified world, a surging, colourful, resounding world just as we speak of the space around us. […] What I am describing here is the ascent to imaginative perception’ (GA 82, 8 April 1922).
The role of the senses
Natural science relies on perception. How could it evolve further so that we learn to create our usual sensory perception from within ourselves and place it outside of us? How can we get to know the sensory world, which usually is a world of effects for us, as the world of our own creation? How can we learn to not only take in truth (‘wahrnehmen’ in German) but create truth (‘wahrschaffen’) and give truth (‘wahrgeben’, terms coined by the German mathematician Ulrich Pinkall) in the way we learn to think mathematically?
In the realm of the higher senses, this creating and giving of truth has long been a principle of civilization: young children first learn to perceive tones, sounds, concepts and the ‘I’ of other people around them. These perceptions inspire them to inwardly take hold of these qualities and actively express them as they learn to sing, speak, think and develop their own ‘I’.
In the realm of the middle senses such activities have so far been largely restricted to the arts. Could it become a basic method of natural science – for instance with the help of ‘symbolizing’ – in the same way as it already is in the realm of the lower senses through mathematical thinking? Children do not yet learn at school to create and give the truth in this realm of the middle senses as they do in mathematics and language.
The fact that, in vision, the complementary colour is created as a sensory imagination shows that there is a latent physiological-astral basis for this ‘truth-giving’ and we can experience it in the afterimage that appears in the complementary colour. But to work with these qualities entirely without any sensory stimulus, as mathematicians do when they work with qualities of movement and balance or as we all do with higher qualities when we speak, is something that has so far not been practised or attempted much. The same is true for taste, smell and warmth.
Imagination
The Mathematical Study Conference, organized by the Section for Mathematics and Astronomy will explore these topics and is explicitly not for mathematicians only. Imagining with the qualities of the lower senses by Charles Gunn and Detlef Hardop: basic mathematical elements are will expressed in thought. Mathematics gets these to flow through willed thinking. The latter aspect will be illuminated by Constanza Kaliks who will use the example of Nicolas of Cusa and by Detlef Hardorp using the example of Johannes Kepler. In his lecture of 10 April 1922 (GA 82) Steiner said, ‘Those who have discovered morality in the world in which mathematics as understood by Novalis is experienced will know that this morality appears in this field, that it appears to someone who is entirely detached from the world of the senses as intuitions that are at the same time inspirations and imaginations.’
In relation to imagining with the qualities of the higher senses, Salvatore Lavecchia will speak about the following meditation by Rudolf Steiner from 1903: ‘Reflect on how the point becomes sphere and yet remains a point. Once you have understood how the infinite sphere is merely a point, return, for then the infinite will appear to you as finite’ (GA 264, letter to Günther Wagner of 24 December 1903).
Ulrich Pinkall will speak about his concepts of ‘truth creating’ and ‘truth giving’ and about the set that contains itself, Andreas Heertsch on the scaling of spiritual experiences. Jutta Nöthiger’s contribution will be on listening and speaking. She will also try to use speech exercises to make the activity within sensory perception tangible.
We will then look at imagining with the qualities of the middle senses. Matthias Rang will look at all the senses as relationships of trust, followed by Ulrich Pinkall, who will discuss the inwardness of light, the circular nature of imaginary numbers and the fact that Rudolf Steiner basically anticipated the Schrödinger equation in his ‘Warmth Course’ (GA 321).