Department of Mathematics
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One of the fundamental problems in Science is to effectively decide when two objects are "same" or are indistinguishable. Felix Klein's famous Erlangen Program established a basic connection between geometries and their automorphism groups and gave rise to the belief that the groups somehow carry the same information as the corresponding geometries. In this talk we show that this is indeed the case with Alfred Tarski's treatment of plane geometry. In modern terminology, we prove that the plane geometry is mutually interpretable within the elementary theory of the group of automorphisms (of its standard model). In particular, we show that points, lines, incidence relations, perpendicularity, ternary relation of betweenness (i.e.T(A,B,C) iff the point "B" is between the two points "A" and "C"), distances etc. are definable strictly within the algebraic language of group theory. Thus while geometry may be thought of as 'visual' and group theory 'abstract', the two theories are indistinguishable apart from our 'figures of speech'.

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