Some people like Peter Norvig prefer top-down, hackers like me and you prefer bottom-up. Many problems can be solved either way. But for some problems, if you use the wrong approach, you're gonna have a bad time. See Ron Jeffries' attempt to solve sudoku.
The top-down (mathematical) approach can also fail, in cases where there's not an existing math solution, or when a perfectly spherical cow isn't an adequate representation of reality. See Minix vs Linux, or OSI vs TCP/IP.
The top-down (mathematical) approach can also fail, in cases where there's not an existing math solution, or when a perfectly spherical cow isn't an adequate representation of reality. See Minix vs Linux, or OSI vs TCP/IP.