Is all mathematics computable?

No matter how good a computer is, it will never compute the whole sequence of PI, but we can approximate it to arbitrary degeree. We can also implement programs that can do calculus and linear algebra. But what about other subjects, Group theory, Topology, ... Is some branches of mathematics more 'computable' than others?

One reason for asking is that 'teaching a machine to do something' also makes me learn it better in the process.

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A computer has a fixed instruction set, so we can always generate problems which it will not be able to solve. – Asaf Karagila Oct 15 '12 at 11:18
@Asaf Without further detail, it is difficult to infer the meaning of that remark. – Bill Dubuque Oct 15 '12 at 13:15

It is certainly the case the some areas of mathematics can benefit more from computers than other areas of mathematics do. Since the question is not very clear I will keep the answer short and just mention that computers are used in mathematics in several ways. One is a computational tool, another is to verify proofs and also to obtain proofs, and yet another use of computers is in simulation. Searching for: automated proof systems will give you a lot of results on at least one way computers are helpful in many areas of pure mathematics.

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