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.
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.
Unfortunately much of classical mathematics is not computable, because of the reliance on the law of excluded middle or the axiom of choice. These principles are inherently opposed to computability, because they allow us to conclude that something "exists" without giving a construction of the object. For example, the axiom of choice allows us to assert that a section exists to any surjection of sets, but does not provide us with any means of constructing such a section.
Constructive mathematics, on the other hand, which uses intuitionist logic rather than classical logic, is entirely computable: from every proof we can extract an algorithm.
There is of course the issue of countable choice, which most constructive analysts accept because it enables them to concluded that the Cauchy reals are sequentially complete. Many constructive mathematicians believe that countable choice has a computational justification, whereas others disagree with the use of this principle.
The development of topos theory and the recent emergence of homotopy type theory have reignited a good amount of interest in constructive mathematics and computer proof assistants, so you might be interested in looking into this.
