> A smooth k-dimensional submanifold of the Euclidean space RN is its subset which in a neighbourhood of its every point is a graph of a smooth mapping of Rk into R(N - k) (where Rk and R(N - k) are coordinate subspaces). This is a straightforward generalization of most common smooth curves on the plane (say, of the circle x2 + y2 = 1) or curves and surfaces in the three-dimensional space.

I just wanted to provide a somewhat humorous convergece between super-abstract mathematics and the concrete mathematics VI Arnold advocated: The category of smooth manifolds is the idempotent completion of the category of open subsets of Euclidean spaces:

http://mathoverflow.net/questions/224569/smooth-manifolds-as...

https://ncatlab.org/nlab/show/smooth+manifold#patching_as_id...

When you unravel the definition you get a very concrete idea of what a manifold is without the bureaucracy of charts and atlases and whatnot. In general, Lawvere's mathematical career is an ambitious program to cast classical mechanics in categorical terms. I see this as the synthesis of the two traditions.

Another way in which abstract mathematics is converging with concrete mathematics is the Curry-Howard-Lambek-... correspondence between logics, programming languages, and category theory. The most abstract branch of mathematics happens to be very amenable to the very physical, concrete process of computation!

I think abstract definitions that are far divorced from the concrete objects they are trying to capture is a deficiency in the abstraction. The correct abstraction somehow miraculously becomes far more concrete than any intermediate definition. For example, homotopy theory has a very ugly definition in set based mathematics, but by abstracting the essentials, homotopy type theory looks very promising at reconciling the bureaucracy of rigor required to do homotopy theory in sets and the very geometric intuition one has for homotopical objects.