Number theory + geometry = philosophy
The motto of this site. Joining analytical thinking in numbers with visual thinking in forms leads to questions that go beyond both.
Number theory + geometry = philosophy. That is the motto of this site. It means bringing together two ways of thinking: the logical-analytical, which thinks in numbers and quantities, and the pictorial-intuitive, which thinks in forms. Together they are meant to produce a whole and deeper understanding. It is about the art of thinking in connections.
Explicitly not entertainment: this is neither “entertainment” nor “edutainment”. Anyone who thinks independently and seeks fundamental truths should find material here, as an impulse to question, think further and develop.
What are numbers?
At the beginning there are questions:
- Do numbers exist independently of us, or only in thought?
- Does the order lie in the numbers, or in the way our mind grasps them?
- Do numbers follow a rule that stands above them, or are they the rule themselves?
- Is this rule the work of a mind?
- And what connects number, space and time?
Numbers seem to us something fluid and fleeting, without substance, as if only our mind put them in order. But draw them as symmetrical figures and their order proves fixed and immovable. It is this order that is meant here by the tetractys.
Why geometry
There are good reasons to look at number theory through the eyes of geometry, more precisely the geometry of circles and spheres, perfect symmetry, including all the symmetrical structures drawn within them: star polygons in the plane and crystal lattices in space. Some of these geometries reproduce the divisibility of the natural numbers one to one. In them one finds ordering structures that can answer open questions of number theory.
The great advantage: a figure knows no counting system. It is not bound to the decimal notation that silently shapes our thinking about numbers. Anyone asked to multiply in their head in another number system notices how deeply the decimal system is rooted in us. The geometry of the simplexes counts in no system, and yet, surprisingly, it “counts on ten fingers”: ten and its multiples appear in it as turning points.
Who thought all this up?
People who come across these connections for the first time often ask: who thought all this up? That is exactly the point. These are timeless laws that can only be recognised. To recognise them, though, one has to study them intensively, and that is where it usually fails. For who can see at a glance that a closer look is worthwhile? Only someone who already has an inkling.
The physicist Hans-Peter Dürr distinguished between knowledge for use and knowledge for orientation. This is about knowledge for orientation. Not about what use something is, but about questions like:
- What is the world?
- How am I embedded in it?
- What is cause, what is effect?
- What is the meaning and purpose of the human being?
There will only ever be a small group of people interested in these questions. To them: Welcome to the adventure of the tetractys, welcome to the adventure of life.
For Pythagoras this philosophical side of number stood above all the natural sciences. The following chapters pursue the questions one has to ask anew once one has recognised the system of the tetractys.