The Birth of Multiplication

Count to ten. You have just performed the operation from which the whole of mathematics arises — and whose inner tension remains unresolved to this day.

This book follows a single movement: how addition arises from counting, multiplication from addition, and how the two move apart without ever fully separating. It presupposes nothing but the willingness to calculate for yourself.

The movement

Counting and addition — One step after another, all steps the same. Brahmagupta’s discovery of zero. The first crack.

Repetition — Four times three is a change of level: it is no longer the numbers that are counted, but the repetitions. The Greek gnomons show it geometrically; the Indian place-value system builds it into every number.

The price of inversion — Every inverse operation costs something. Subtraction costs nothing essential. Division destroys discreteness. Roots break countability and finally order.

The additive world is richer than expected — Fibonacci produces the golden ratio, whose determination is multiplicative. Pascal’s triangle, built purely additively, contains a complete test for primality.

Emancipation — Neighbouring numbers are multiplicatively foreign to each other. The space of primes has infinitely many dimensions. The periods grow explosively.

The tension finds form — The Euler product proves that the sum over all natural numbers and the product over all primes have the same value. And Riemann’s question is whether this tension has a perfect harmony.

What is special

The text consistently distinguishes what is theorem, what is interpretation and what is numerical evidence — a discipline rarely found in popular mathematics books and almost never in philosophical ones.

It does not calculate for you but has you calculate: the experiment boxes are not exercises, but the points at which it is decided whether what is claimed is right.

And it ends with a proposal rather than an assertion — a guiding concept that makes the open questions of number theory readable as variations on a single theme:

From the same arises the different — irreversibly. Iteration that observes itself produces structures that it can neither control nor undo.

Relation to the other books

The same movement that is followed here in numbers is developed by Hegel’s logic as a general theory of categories: that quantitative iteration leads to qualitative transitions and that every determination has its price. Hegel’s Logic – Logic of Thinking presents this conceptually; this book presents it in an object that cannot be talked round.

Readable on its own — no prior knowledge beyond school arithmetic is needed.

Bibliography

Year

2026

Edition

Kai Froeb Edition

Licence

Copyright

Where to buy

Download available soon.

More by this author →