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.

Sacred mathematics

For the philosophers of antiquity, mathematics and geometry were a way of raising the eye of the soul to the divine. Plato praises geometry as knowledge of what always is, not of what comes to be and passes away in time. Its real use, he writes in the Republic, is hard to believe:

in each of us there is an eye of the soul which, when by other pursuits lost and dimmed, is by these purified and re-illumined, and is more precious far than ten thousand bodily eyes, for by it alone is truth seen. Plato, Republic 527d–e, translated by Benjamin Jowett

Only in this sense can the saying be understood that is said to have stood over the entrance of Plato’s Academy: “Let no one ignorant of geometry enter.” It is recorded, however, only by commentators of the 6th century, some 900 years after Plato. The study of this sacred mathematics led, through contemplating perfect figures and numerical ratios, into the depths of one’s own being.

Johannes Kepler went further. God gave human beings a mind not only to earn their living, which many animals do more skilfully, but to advance from the visible to the causes of being and becoming, even when no use comes of it. And geometry, Kepler says in the Harmonices Mundi of 1619, existed before the creation of things, eternal like the mind of God himself, and supplied him with the archetypes for creating the world. Kepler had himself portrayed with a pair of compasses in his hand.

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.

Sense and nonsense of number mysticism

Before asking such questions one must settle what is meant by “numbers”. Numerals are invented symbols for quantities, and they depend on a place-value system, in our case the decimal system. It is a freely chosen order that makes arithmetic easier and large numbers manageable at all. Whoever wants to explore the nature of number must lead the symbols back to their natural state: to sets of points, to the point as unity, the line, the polygon, the solid, the leaps of dimension that the tetractys also shows. Only from geometry can the “number mysticism” of the Pythagoreans, Platonists and Kabbalists be explained.

The holy grail of number theory is the primes. Thirteen points are a prime set, whatever sign they are written with. In another counting system 13 looks different, but the set stays prime. That is why the nature of the primes can only show itself in geometry, and that is exactly what number mystics without knowledge of geometry have failed to understand. One can only reflect on the questions above once one knows the geometry behind the symbols.

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.

Symmetry and asymmetry

A philosophical reading of the tetractys might begin something like this. In physics and biology, symmetry simply means balance. Without it there would be no safe nursery for any development. But chaos is indispensable too: without asymmetry no change, and small errors in the system, mutations, drive development upwards.

The same pair appears in the number line of figures. The combinatorics of 1, 2, 3 and 4 creates a clearly structured basic order to infinity, as it were the one primordial God. The following primes, starting with 5, each leave their tracks in a regular but individual rhythm. As these different rhythms, the divisors of every quantity, overlap, a creative chaos arises that still remains governed by the frame of 1, 2, 3 and 4, by the strict order of 3 and 4, half circle and full circle. The creative male and the preserving female, symmetry and asymmetry complement each other.

Some thoughts still waiting to be worked out: the axes of symmetry always belong to quantities divisible by three. Three is the balancing principle between opposing forces and corresponds in a sense to the zero point, the point of coordination. Circumference and diameter, rotation (the quantity) and movement towards the centre (the divisor), in the widest sense cause and effect, active and passive, swap roles at the “zero crossing”, at quotient 1, and turn into their opposites (see The lambdoma).

What did the ancients know?

A major topic is the lambdoma, a coordinate system whose two axes carry the numbers to infinity and which, set against each other, produces an “irregular regularity” (see The lambdoma). In number theory and geometry it corresponds to the lambdoma of harmonics, and it is the interface between the blueprint of numbers and the world we experience. These strict correspondences of number and symmetry give a firm hold, a philosophical foundation.

A second thought may be speculative, but it would be a pity to keep it quiet: the relations of number and figure mirror the Western teaching of creation and fall with astonishing precision. Creation in six days corresponds to the rhythm of six of the primes, the fall to the disorder that begins right after 24, after the throne of God with its 24 elders (see The throne of God and the number 24). Hexagram and pentagram, too, stand where their hermetic meaning would put them. Above all the pentagram, 5 : 2 = 2.5, and the double pentagram, 10 : 4 = 2.5, in the disorder of the primes and their kabbalistic correspondence with matter raise the question: what did the ancients really know?

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.