[FOM] non-Euclidean geometry and FOM programs

andre rodin andre.rodin at ens.fr
Sat Nov 10 07:47:36 EST 2007

Antonino Drago:

> In fact never Lobachevsky wrote "There exist two parallels",
> rather (I mark
> by two asterisks each negative word)
> "The... assumption can likewise admitted *without* leading top any
> *contradiction* in the results.." (the end of proposition no. 22 in
> Geometrical Studies..., an appendix to R. Bonola: Non-Euclidean Geometry,
> Dover, 1955)
> "1° Dans la théorie *rien* *ne s'oppose* à admettre que la somme des angles
> d'un triangle rectiligne soit moindre que deux angles droits" 2° Dans
> l'hypotèse de la somme des angles d'un triangle moindre que deux angles
> droits, les équations (13) peuvent etre substituées aux équations
> ordinaires
> (15) *sans* mener jamais à quelques résultats *absurdes*....."(Géométrie
> Imaginaire, J. Crelle, 17 1837, p. 302)

I must confess I'm still unable to grasp your point. Lobachevsky's position
seems to be this: the Euclidean axiom of parallels can be wrong for the
physical space and (hence) for an abstract geometrical space. All his
reasoning involves explicit geometrical constructions with multiple
parallels to any given straight line. (Actually Lobachevsky's notion of
being parallel is more specific than "having no intersection point"; given
that there is *exactly* two parallels to any given straight line in his
Thess explicit constructions are certainly NOT  Euclidean models of
Lobachevsky's geometry like Beltrami's model. Lobachevsky didn'n need this
kind of model because he didn't try to reduce his new notion of geometrical
space to the Euclidean one;  instead he tried to work with this generalised
space just like people traditionally worked with Euclidean space, that is,
constructively and relying on geometrical intuition. So he needed to enlarge
the Euclidean intuition and in my view he did this quite successfully. I
suggest to anyone to try to follow Lobachevsky's own method rather than
conceive of his geometry only through Beltrami's or other models;
commentaries to Lobachevsky's Collected Works in Russian by V.F. Kagan as
well as Kagan's books on Lobachevsky's geometry can be of great help for it.
Basic "hyperbolic intuitions" are not so difficult to develop as one can
probably imagine - and it is a funny experience in any event. Lobachevsky
relies on this kind of intuition (which comprises the Euclidean one as a
limit case) and use drawings to support this intuition throughout his works
(his conventions about how to draw hyperbolic objects are also instructive).
So Lobachevsky's approach remains classical except that he works with a more
general notion of space than the Euclidean one. I cannot see where his
reasoning clashes with Classical logic any more than the Euclidean
reasoning. Lobachevsky's geometry in the author's version is a constructive
theory just like traditional Euclidean geometry.  Consider also the

There is a rather widespread opinion about Lobachevsky's work according to
which he allegedly developed a kind of formal theory but didn't provided a
model for it. The above remarks explain why this view is wrong. Actually
looking at Lobachevsky's work from the Hilbertian perspective (formal theory
+ its models) one discovers something tricky: Lobachevsky didn't model his
space in the Euclidean one but made it the other way round. Namely he found
on Lobachevsky's plane (what we now call a model of) Euclidean line (which
he called "oricircle" and found in Lobachevsky's 3D space (what we now call
a model of) Euclidean plane (which he called "orisphere"). These models are
much nicer than Beltrami's since they don't have singularities just like
models of spheric (sometimes referred to as Riemanean) geometry in Euclidean
space. Lobachevsky saw this analogy (which today we could describe as
duality) with the spheric case clearly. This was a crucial observation
because it allowed Lobachevsky to develop the machinery of (what we call)
hyperbolic trigonometry and apply in his proposed geometry powerful analytic
methods. Hyperbolic intuition discussed above was important for
Lobachevsky's work; this issue is also important a philosophical analysis of
his work. But to recognise Lobachevsky's achievements from a narrow
mathematical viewpoint one is not obliged to go for it. Lobachevsky's
contemporaries who didn't take seriously his notion of "Imaginary geometry"
couldn't deny the power of his new analytic methods (which among other
things allowed for calculating of new types of integrals). I guess (albeit I
have no a direct evidence) that quite a few mathematicians of his time would
consider the "Imaginary geometry" as nothing but a pardonable excuse for the
invention of the new analytic tools.

I which everybody a good weekend,

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