I disagree with the premise of this question.
Conventional computers follow a program written by a human. I think, for example, Daniel Moskovich's answer about simplicial sets is something that a human programming a computer (or a computer scientist) would think of when trying to program a computer.
Formalisms like these are things that we humans think of when programming a computer. Hence we have a tendency to think of the as "more mechanical", or "more like a computer", etc.; but I think it's a mistake to think that this is something that a computer "would come up with" on its own. Really it's us humans that come up with them, just when we are thinking in terms of computing.
There are computers which can be thought of as actually "thinking" in a way similar to humans (as opposed to just following a program), e.g. IBM's Watson computer. They need some large data set to learn from, though (just like we do), and if this large data set is all of the mathematics created by humans, then I think the mathematics produced by the computer would look a lot like things "a human would think of"!