This post offer some argument in favor of the post by danseetea, yet it might be a bit too long for a comment.

Let me try to propose some candidate for the interpretation of what does it means by "top down" and "bottom up".

I think a vague definition of mathematics = study of interesting structures. I believe we generally agree that mathematics is the study of structures, the difficulty lies in trying to make clear what is interesting.

A. top down approach: Give a suggestion of what is interesting with the risk of overkilling. Candidates:

Set theory: Offer none explanation what is interesting. (Failed)

Category theory: Interesting properties are properties invariant under certain class of function. That seems better but possibly still leave some junks inside.

B. bottom up approach : Try to find some generator of the set of interesting structure with the risk of leaving out a few things. Candidates:

Number theory (Failed) It seems at least we need something like geometry.

Number theory + Geometry

Number theory + Combinatorics + Topology

Analysis +Algebra + Geometry

I want to fail the the last two candidates since it appears to me that Topology and Algebra seem to contain certain elements of the top down (mid air?) approach rather than bottom up. That is because not all algebraic structures and all topological structures are interesting. Certain topological results are interesting perhaps because they describe properties of interesting mathematical structures. There are also pathological results but am I right to say that mathematician generally has limited interest in them?
 
Number theory, geometry seems to be good since they come from time and space which are the two basic elements of our perception of the external world.