Why are there mathematicians that do not use computers? I was watching a video on Andrew Wiles and his proof of Fermat's Last Theorem and I quite liked the video, especially the complexity of the proof only to prove a simple concept which can be understood by most people.
I also liked the graphics they used to illustrate elliptic curves and modular forms.
But then Andrew Wiles said that he never uses a computer, he only uses pen and paper and I also heard of other mathematicians that don't use computers.
Do they not use computers because there are problems only a mathematicians can solve or are there other motives? Wouldn't the proof have taken him less time if he used a computer to assist him?
 A: I work in software and I have an amateur interest in mathematics, and from what I can tell many theoretical mathematicians don't have much use for computers because of the domains they work in. In order for a mathematician to utilize a computer, the problem must either be something that requires a lot of computation or is already formalized enough for a proof assistant or theorem prover to attack.
If you need one closed form solution to an easy differential equation, just using the stuff you learned in calculus is easier than figuring out how to use software that does integration for you, much as you wouldn't turn to a calculator to multiply six times nine. Computers can be useful for areas of mathematics where you need to come up with lots of closed form solutions (computer algebra systems) numerical integration or integral transforms, or running other intensive computing tasks in other domains like computing the class number (or other properties) for a large number of number fields.
Many of these problems are tedious and not particularly interesting to theorists, even if they're quite useful for applied math; Further, solving the problem often requires writing software to do it, given theorists often work in unexplored areas. For theorists, you often want to prove a statement of some sort, and that requires a proof assistant of some sort; While they can be useful, the foundations for most graduate level mathematics aren't formalized in first order logic, and where they are they're usually inaccessibly unwieldy compared to more informal reasoning.
Further complicating the picture is the traditional method of formalizing mathematics is built on the foundation of set theory in first order logic, which is often incompatible (or at least unwieldy) with metamathimatical reasoning about categories, nonstandard models, and other foundational issues. So if mathematicians who worked in this field tried to use computers, they'd spend more time writing software and formalizing existing math than doing new work.
As an example, very little of Wiles proof has been formalized in a form that can be verified by mechanized reasoning, because most of the branches of mathematics that it rests on have yet to be formalized. This may change in coming decades, as theorem provers and proof assistants get more advanced, but for the time being computers are useful for the most mature, formalized areas of mathematics that is largely the domain of physics and applied math.
