On this understanding, a Turing machine is not a computer, but a mathematical abstraction of a certain kind of computer. Computer science recognises many kinds of computer and computation, and we should try to include them all. But according to the above, this would make Turing machines computers, thereby contradicting the previous claim.
Computers contain and manipulate symbols.
To a philosopher this is one of the most controversial claims and to a computer scientist one of the most obvious. This contrast needs to be accounted for. For example, one can say that computation has access only to the forms of these symbols, which are therefore mere patterns, and have no meaning to the computer. One approach to describing a computer is as something whose physical or causal states mirror the state-transitions of a suitable abstract machine, defined mathematically. Many elaborations on this theme have been given, but I find them all unsatisfactory, for a number of reasons.