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Or maybe also a combination of the two, but not something else, whether numeric, nor symbolic. Do they cover the whole field?

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No. You could make a decent argument that every algorithm is in some sense a symbolic manipulation, though not in the sense normally used for symbolic computation - they are not explicitly algebraic, and more naturally find expression as logic and set theory (unsurprisingly). Taking the wider view of symbolic manipulation though, even numerical analysis is just symbolic calculation, it's only at the external semantic level that we recognise it as an approximation of what we actually want to do. The computer is still just pushing bits around.

However Computer Science extends beyond algorithms, even though they are central to CS. It includes complexity, computability, plus practical subfields like networking, machine intelligence, compiler and language design &c. which involve more than just designing algorithms.

Just to pick a few illustrations of this: designing a programming language is not a computation of any sort (the compiler is of course), but it is certainly computer science; the construction of a complexity reduction is not a computation (but the reduction itself is; the choice of processor layout in parallel computing is a choice made by the designer.

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