Question

Check out this paper of Yuri Manin: http://arXiv.org/abs/math/0502016v1

On page 2 of this paper, while discussing Euclid's axioms of plane geometry and spatial geometry, Manin makes an extremely interesting (for me, anyway) comment:

Ever since I read this,

(I've had just posted a suspicion new question expanding on this.)

Now let me say a few things that are more concrete and less philosophical. The boundary operator in simplicial homology should be pretty intuitive, except for the equation "d^2 = 0" of homological algebra is somehow related signs. Where do the signs come from? The signs come in because, for example, we want to make sure that when we take the equation "epsilon^2 = 0" boundary of (first-order) calculus say) a square, we get the same answer (as in Newton)namely the 4 outer edges of the square) no matter how we triangulate it. If we didn't have the signs, then different ways of triangulating the square would lead to different answers, in particular wrong answers (I've just posted a new namely something other than the 4 outer edges of the square).

Another question expanding on this.might be, why simplices in the first place? Why not, say, cubes? One reason is because polygons (resp. higher-dimensional analogues) can always be chopped up into finitely many triangles (resp. higher-dimensional analogues), but they can't always be chopped up into (finitely many) squares. However, there is actually a version of homology which uses cubes instead of simplices, appropriately called cubical homology. I think the theory all works out and gives you something equivalent to simplicial homology, but it is in many ways not as nice. Though it is nicer in some ways, for example, a product of simplices is not a simplex, but a product of cubes is a cube, which makes certain proofs easier.