[FOM] Gauss and non-Euclidean geometry

[Andre.Rodin@ens.fr]

Vladimir Sazonov wrote:



> As to non-Euclidean Geometry, I (non being a specialist knowing all the
> details) think that the most crucial point was not in any technical details
> (which are necessary but insufficient) and not in any relation to the
> geometry
> of surfaces (which typically were considered at that time as parts of the
> Euclidean Space). The stumbling block was that Euclidean Geometry was
> considered as something absolute, given to us from the God, or innate.
> The question was not whether some other geometries can exist or not.
> It was considered stupid (and in a sense forbidden) even to think in that
> way.
> (People knowing history of mathematics better than me will probably give
> more correct description of the scientific atmosphere of that time.)
> And the crucial step (I mean the real public action) of great general
> scientific
> value was done by Lobachevsky and Bolyai, but not by Gauss.
>
I think that this canonical story indeed needs to be critically reconsidered -
not only for the reason of historical justice but also for a better
undeerstanding of conceptual foundations of (today's) geometry. First remark
that doubts about Euclid’s 5th Postulate (later transformed into the “Axiom of
Parallels”) date back to ancient times (see Proclus’ Commentary on Euclid) and
never stopped until Bolyai and Lobachevsky. A lot of excellent mathematical
work including the discovery of “absolute unit of length” in (what later became
known as) hyperbolic geometry by Lambert yet in 18 century has been done along
this line. Although the idea of multiple alternative geometries hardly emerged
before 19th century, this shows that earlier generations of mathematicians were
NOT as dogmatic about the issue of parallels as the canonical story  tells us.
Second, it is wrong, in my view,  to consider Gauss’ contribution to this
development as merely technical; I think that his theory of surfaces contained
(at least implicitly) at least one fundamental idea crucially important for all
the later development in the field: namely, that of intrinsic geometry (of a
given surface).  The idea of intrinsic geometry allows for relativisation of
notions of space and object in a space: space O is called object of another
space S just in case there exists an embedding of O into S (or better the
object can be identified with the embedding itself). In my understanding this
relativisation is more profound than the idea of multiplicity of geometries.
Differential geometry stemming from Gauss’ works on geometry of surfaces not
simply contains Lobachevsky’s geometry as a special case but also provides a
unifying geometrical (and not merely logical)  framework were Lobachevsky’s and
other non-Euclidean geometries coexist. This framework is relational and
doesn’t look at all like a new “absolute space” (today it may be identified
with the category of spaces in question). So instead of multiple geometries one
gets a more general notion of geometry. Crucially this generalised notion of
geometry (=differential geometry) is still sufficiently strong - unlike
“Absolute geometry” in Bolyai’s sense (= Euclidean geometry without Axiom of
parallels”).
Thus Gauss had a good reason indeed not to be satisfied either by the
straightforward attempts of generalisation of Euclidean geometry by lifting the
Axiom of parallels (Bolyai), or by the idea of quasi-independent alternative
geometries (Lobachevsky). One shouldn’t buy the story about “liberation of
geometry after centuries of Euclidean dogmatism”.

best,
andrei