Only a bad Bourbaki scholar may guess that geometrical pictures of numbers must not be a theme of pure mathematics. Essential steps of mathematical progress can be seen as fruits of quarrels between the theory of numbers and the concept of geometry. The form of Euclid's Elements is essentially conditioned by the difficulty that the diagonal of a unit-square can only be measured by √2. Modern real analysis was only possible after Descartes' idea of analytic geometry. The understanding of mathematics as a theory of structures is founded on the discovery of non-Euclidean geometries. So I want to clarify the concept and the position of Natural Geometry in answering two fundamental questions:

1. Does natural geometry of numbers be a non-Euclidean geometry?

  • YES
    • This geometry can start without using Pythagoras' theorem.
    • In the beginning this geometry does not possess Euclidean straight lines and lengths on these lines.

  • NO
    • The sum of angles in conformal triangles is equal to two right angles, as in Euclidean triangles.
    • Natural geometry generalizes the Euclidean concept of similarity (Ähnlichkeit) and shares this idea with Euclidean geometry and only with this geometry.

2. Does natural geometry of numbers be a Non-Cartesian geometry?

  • YES
    • This geometry starts without using Cartesian coordinate systems.
    • In the beginning a number is not separated in its real and imaginary parts. Coordinates are not generally used to describe the imaginary space-parts.

  • NO
    • Natural geometry uses numbers to describe geometrical figures. It uses this basic idea of Descartes.
    • Natural geometry looks out for a perfection of Descartes' basic idea by replacing real numbers with complex ones, by 'unification' of the theory of numbers and the geometry of Euclidean triangles, in seeing the geometric world as H-number-line.


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