I intend to write in more detail on this topic. For now, here is a brief summary.

The development of non-Euclidean geometry caused a profound revolution, not just in mathematics, but in science and philosophy as well.

The philosophical importance of non-Euclidean geometry was that it greatly clarified the relationship between mathematics, science and observation. Before hyperbolic geometry was discovered, it was thought to be completely obvious that Euclidean geometry correctly described physical space, and attempts were even made, by Kant and others, to show that this was necessarily true. Gauss was one of the first to understand that the truth or otherwise of Euclidean geometry was a matter to be determined by experiment, and he even went so far as to measure the angles of the triangle formed by three mountain peaks to see whether they added to 180. (Because of experimental error, the result was inconclusive.) Our present-day understanding of models of axioms, relative consistency and so on can all be traced back to this development, as can the separation of mathematics from science.

The scientific importance is that it paved the way for
Riemannian geometry, which in turn paved the way for Einstein's
General Theory of Relativity. After Gauss, it was still reasonable
to think that, although Euclidean geometry was not * necessarily
* true (in the logical sense) it was still * empirically *
true: after all, draw a triangle, cut it up and put the angles
together and they will form a straight line. After Einstein, even
this belief had to be abandoned, and it is now known that Euclidean
geometry is only an approximation to the geometry of actual,
physical space. This approximation is pretty good for everyday
purposes, but would give bad answers if you happened to be near
a black hole, for example.

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