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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.2012-10-15
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    @Asaf Without further detail, it is difficult to infer the meaning of that remark.2012-10-15

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