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I’m referring, naturally, to Felix Klein and his Erlanger Programm (http://en.wikipedia.org/wiki/Erlangen_program). Colloquially put, “groups rule geometry OK”. To be more sophisticated about it, geometric languages are related by a type of Galois connection to the subgroup structure of a “master” group, which for Klein’s purposes could normally be thought of as the complex projective linear group. Euclidean geometry, for example, has a smaller group but a richer geometric language; complex projective geometry has fewer theorems but they are more powerful, for example in the way that parallel lines need not be exceptions, and conics intersect also in a way that can be described much more simply.

Klein’s work, once a revelation, became later a Procrustean bed: a not uncommon fate for big ideas in mathematics. I was thinking about this just now from the aspect of pedagogy in geometry: Klein was interested in this aspect, I was brought up on “transformation geometry” myself, and read Coxeter’s “Introduction to Geometry” at about 17 which is a serious implementation. I don’t regret the transformation geometry. But I would feel that a doctrinaire view that “Klein was right” would be rather jarring.

Which of the following ring true, then?

(a) Lifting up from the homogeneous space to the group is usually progress in geometry.

(b) Different ways to represent a homogeneous space as coset space represent opportunities.

(c) There is a “theory of the classical groups” and it still informs our view of geometry.

(d) Despite Euclid, the axiomatic method is really no more significant now in geometry than in any other subfield of mathematics.

(e) G-structures on manifolds are a more inclusive way to relate geometry to Lie groups.

(f) In fact structures on manifolds in general, and the pseudogroup concept, are a better explanation of geometry.

(g) The privilege attached to the projective group is now misleading, since the general linear group is the correct starting point. The privilege attached to invariance/invariants is also a partial view.

(h) Category theory is always going to unify more than group theory does, and we are still working through the consequences.

(i) In particular topology has taught us about much more than the rubber sheet, and algebraic geometry that we didn’t know what a point was.

More attitudes could be added here. But is the following the correct one: the criterion for a successful programmatic approach in mathematics is more than just being a philosophical “moment”? And, in that light, Klein actually did fall short?

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This could make an interesting blog post. –  Mariano Suárez-Alvarez Mar 28 '11 at 14:33
The Klein's view of geometry fits naturally into a more generalized one (due to Cartan), that includes not only "homogeneus" geometries but also "lumpy" ones such as Riemannian geometry: books.google.com/… –  Qfwfq Mar 28 '11 at 15:05
Well, yes, a G-structure for the orthogonal group. –  Charles Matthews Mar 28 '11 at 15:28
When you say "in particular" in part (i), what statement is being specialized? –  S. Carnahan Mar 28 '11 at 15:55
Oh, I meant in relation to (h): functors are the standard method of algebraic topology, which is not just about homeomorphism invariants ("rubber sheet geometry" as the older books said); and "point" in scheme theory became a way of thinking about the Yoneda lemma. –  Charles Matthews Mar 28 '11 at 15:57

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