[FOM] non-Euclidean geometry and FOM programs
drago at unina.it
Thu Nov 15 19:37:41 EST 2007
November 10, 2007 andre rodin" <andre.rodin at ens.fr wrote:
>All his [Lobachevsky] reasoning involves explicit geometrical constructions
>parallels to any given straight line.
>These explicit constructions are certainly NOT Euclidean models of
>Lobachevsky's geometry like Beltrami's model.
This idea does not appear in Lobachevsky writings. Only in Geoemetrische
Untersuchungen... (1840) he devotes the proposition no. 23 to a "geometrical
construction" for finding out the parallel line to a given line (actually,
its proof is incorrect, because it relies upon a wishfull sentence: "it must
exist such a parallel line FG").
Instead, Lobachevsky's way of basing hyperbolic geometry was either to argue
upon the sum of the angles of a triangle (in the above quoted writing) or to
exploit the isomorphism between Euclidean spherical trigonometry with
hyperbolic spherical trigonometry, except for the argument of the functions
which in the latter case is an imaginary argument..
Hence, no models. This notion was invented in a time in which mathematicians
thinked that geometry must represent the real world; the models persuaded
them that non-Euclidean geometries too belong to the real space.
After Hilbert the notion of a model took a different meaning.
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