[FOM] non-Euclidean geometry and FOM programs
andre.rodin at ens.fr
Mon Nov 19 07:49:24 EST 2007
andre rodin" wrote:
>> All his [Lobachevsky] reasoning involves explicit geometrical constructions
>> with multiple
>> parallels to any given straight line.
>> These explicit constructions are certainly NOT Euclidean models of
>> Lobachevsky's geometry like Beltrami's model.
Antonio Drago wrote:
> 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").
I refer to "New Foundations of Geometry with the Complete theory of
Parallels" first published in Russian in Kazan in 1835-1838 (then
republished in 1949 as a part of Complete Works but for the best of my
knowledge never fully translated in another language). The "New Foundations"
is the most systematic presentation of Lobachevsky's work. Unfortunately I
don't have this text at hand and cannot check. But do you take it into
consideration when you claim that "the idea doesn't appear in L. writings"?
> Hence, no models.
Agree, but nothing like formal theory (or its prototype) either!
>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.
Disagree. Lobachevsky believed that the real space could be non-Euclidean
without using the notion of model. Beltrami used what we now call a model
trying to show that Lobachevsky's geometry was "really" about the
pseudo-shpere, that is, for saving the traditional Euclidean viewpoint. (He
wasn't quite satisfied by this reduction since it didn't work in the 3D
case. In the second paper of 1868 he found a more satisfactory solution
identifying Lobachevsky's spaces as Riemanean manifolds of constant negative
>After Hilbert the notion of a model took a different meaning.
It is plausible but I'm not quite convinced because I cannot clearly see
what the pre-Hilbertian notion of model might be. Something like
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