What is the historical importance of non-Euclidean geometry?Gowers is a brilliant mathematician, but this misses a few points.

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.

Gauss applied spherical geometry to the surface of the Earth, so he knew of scientific importance for non-Euclidean geometry in the early 1800s.

The first big application of non-Euclidean geometry to physics was special relativity, not general relativity. The essence of the theory developed by Poincare in 1905 and Minkowski in 1907 was to put on non-Euclidean geometry on 4-dimensional spacetime. It was defined by the metric, symmetry group, world lines, and covariant tensors. Relations to hyperbolic geometry were discovered in 1910. See here and here. Later it was noticed (by H. Weyl, I think) that electromagnetism could also be interpreted as a non-Euclidean geometry (ie, gauge theory).

Einstein missed all of this, and was still refusing to accept it decades later.

Yes, general relativity was a great application of Riemannian geometry, and yes, it comes in handy if you are near a black hole. But the non-Euclidean geometry of special relativity has influenced most of XX century physics. It was earlier, more important, and more profound. That is what the mathematicians should celebrate.

It is especially disappointing to see mathematicians get this history wrong. Most physicists do not have an appreciation of what geometry is all about, and physics textbooks don't necessarily explain that special relativity is all a consequence of a non-Euclidean geometry. But the geometry is right there in the original papers by Poincare and Minkowski on the subject. Most mathematicians probably think that Einstein introduced geometry to physics, and therefore credit his as a great genius, but he almost nothing to do with it.

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