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SUMMARY:Victor Pambuccian (Arizona State University\, USA)
DTSTART:20210417T140000Z
DTEND:20210417T150000Z
DTSTAMP:20260423T021216Z
UID:YMC/23
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/YMC/23/">Axi
 omatic problems in ordered geometry and in the arithmetic of the even and 
 the odd</a>\nby Victor Pambuccian (Arizona State University\, USA) as part
  of Yerevan Mathematical Colloquium\n\n\nAbstract\nIn 1882\, Moritz Pasch 
 axiomatized ordered geometry\, the geometry of betweenness. One of the axi
 oms 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 p
 ossible way to axiomatize ordered geometry. In other words\, can one axiom
 atize ordered geometry with statement about no more than 5 points?\n\n T
 he second part of the presentation will focus on the question why Theodoru
 s of Cyrene stopped in his presentation of the irrationalities of square r
 oots at 17\, as Plato lets us know in his dialogue Theaetetus. According t
 o the interpretation of Jean Itard\, amplified later by Wilbur Richard Kno
 rr\, 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 severa
 l 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 i
 rrationaility of the square root of 17 is unprovable\, while in a richer a
 rithmetic of the even and the odd this is still an open problem\, the olde
 st unsolved problem inherited from the ancient Greeks.\n\n If time permi
 ts\, I will look at two proofs of the fact that 30 is the largest number a
 ll of whose totitives (numbers less than itself and relatively prime with 
 itself) are prime numbers (1 is considered a prime number in this statemen
 t)\, one of which was claimed to be simpler and try to make that statement
  of simplicity precise.\n
LOCATION:https://researchseminars.org/talk/YMC/23/
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