Even Kurt Gödel, who's incompleteness theorems are perhaps the most well-known examples of the limitations of mathematics, is widely regarded as a Platonist. He, like many mathematicians, regarded mathematics as more real than the physical world. For a Platonist, theorems are timeless and eternal. Mathematicians' role is to discover and document them as purely as possible. Paul Erdős expressed this sentiment by imagining that the most beautiful proofs came from a book written by God.
On the other hand, few would claim that Linux existed before Linus Torvalds started writing it in 1991. Even a software engineering concept like structured programming is usually described as being founded by Edsger Dijkstra, even though the mathematical theorem that underpins the movement could be said to have been discovered (by Corrado Böhm and Giuseppe Jacopini).
Some mathematicians do leave room for authorship in their understanding of their profession. Leopold Kronecker once said that Bertrand Russell went further and said that integers were also created by man - or at least they could be constructed using mathematical logic.
The defining characteristic of authorship (as opposed to invention) is that the subjectivity of the author is imprinted on the work. One example of this in mathematics is the calculus. Isaac Newton and Gottfried Leibniz both discovered the calculus, but they approached it in different ways. I would argue that their divergent expressions of the same idea are best understood through the lens of authorship, especially given the importance Leibniz placed on notation and presenting his thoughts for human understanding.
But by and large, mathematicians are better described by Roland Barthes' account of tellers of tales before modern authorship was invented:
In ethnographic societies the responsibility for a narrative is never assumed by a person but by a mediator, shaman or relator whose ‘performance’ — the mastery of the narrative code —may possibly be admired but never his ‘genius’. The author is a modern figure.Mathematicians attach their names to their work, but more in the spirit of explorers naming newly discovered peaks than authors cultivating writing credits. It is this emphasis on discovery rather than creation that most clearly differentiates mathematical practice from programming and which means that a purely mathematical education is not sufficient to understand software development.