What modern version control systems are
If you read about new version control systems these days, it's very common to see them put forward as essentially the expression or manifestation of mathematics. Maybe it's graph theory, maybe it's patch theory, but the basic idea is that you build up some formal model of patching or version control and then build a VCS system that implements it. This is not restricted to recent VCSes, either; version control as a whole has long had a focus on formally correct operations (and on avoiding operations that were not formally correct).
It is my new belief that this is a terrible misunderstanding of the true role of a VCS, or at least a usable VCS that is intended for general use. Put simply, in practice a VCS is the user interface to the formal mathematics of version control, not the actual embodiment of those mathematics. The job of a good VCS is to sit between the fallible, normal user (who does not operate in the domain of formal math) and the underlying formal math, working away to convert what the user does to the math and what the math says to what the user can understand and use.
As a user interface, a VCS must live in the squishy world of human factors, not the pure world of mathematics. That's its job; it's there to make the mathematics widely usable. This is going to frequently mean 'compromising' that mathematical purity, by which we really mean 'translating what the user wants to do into good mathematics'. I put 'compromise' in quotes here because this is only a compromise if you really think that the user should always directly express correct mathematics.
(We know for sure that users will not always do so, so the only way to pretend otherwise is to spit out error messages any time what the user attempts to do is incorrect mathematics (including the error message of 'no such operation').)
Does this mean that the mathematics is unimportant? Not at all, any more than your skeleton is unimportant in determining your shape. The underlying mathematics can and should shape the user experience that the VCS puts forward (and so different formal models of version control will produce VCSes with different feels). After all, one job of a UI is to steer users into doing the right thing by making it the easy default, and the 'right thing' here is partly determined by the specific math.
PS: The exception to this view of VCSes is a VCS written as an academic exercise to prove that a particular set of version control mathematics can actually be implemented and work. This software is no more intended (or suitable) for general use than any other software from academic research.

