Axiomatic problems in ordered geometry and in the arithmetic of the even and the odd

Abstract: In 1882, Moritz Pasch axiomatized ordered geometry, the geometry of betweenness. One of the axioms he proposed was the Pasch axiom. We will look at three versions of the Pasch axiom, one of which implies the two-dimensionality of the space, while the other two allow any higher dimension, and will ask whether the original Pasch axiom, which is a 6-variable statement, is the simplest possible way to axiomatize ordered geometry. In other words, can one axiomatize ordered geometry with statement about no more than 5 points?

 The second part of the presentation will focus on the question why Theodorus of Cyrene stopped in his presentation of the irrationalities of square roots at 17, as Plato lets us know in his dialogue Theaetetus. According to the interpretation of Jean Itard, amplified later by Wilbur Richard Knorr, this happened because the method of proof was based on the arithmetic of the even and the odd. To make this statement exact, we present several axiomatizations of what can be called the arithmetic of the even and the odd, and show that, in one such axiomatization one can prove that the irrationaility of the square root of 17 is unprovable, while in a richer arithmetic of the even and the odd this is still an open problem, the oldest unsolved problem inherited from the ancient Greeks.

 If time permits, I will look at two proofs of the fact that 30 is the largest number all of whose totitives (numbers less than itself and relatively prime with itself) are prime numbers (1 is considered a prime number in this statement), one of which was claimed to be simpler and try to make that statement of simplicity precise.

ArmenianMathematics

Audience: general audience

( slides | video )

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Yerevan Mathematical Colloquium

Series comments: "Yerevan Mathematical Colloquium" invites survey talks aimed at a general mathematical audience, that emphasize proof methods, relations between branches of mathematics, possible applications, and open problems.

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