There is a field concerned with mathematically proving the correctness of software. It's only feasible for somewhat small scale things I think, and only worth it for a few % of software tops.
But I interpret the question as asking whether we'll write some bug free software, and I'd say yes. If you mean all or any software, then no.
If I may take this opportunity to rant a little, I think most of us aren't trying nearly hard enough to write bug free software. Myself included. There are many fascinating new developments to be sure, but there are even more already proven tools that are decades old. Yet we're ignoring all of it and writing Javascript. Not really the fast lane to bugfree software. I mean, null was solved well before the start of the century, and most developers still haven't caught on.