[jcranmer]

In mathematics, you need to assert a set of axioms (or preconditions) under which the theorem is held to be true. These axioms can be challenging to figure out; naïve set theory was destroyed by Russel's paradox. Rather famously, the axiom of choice is equivalent (in the sense that assuming one, one can prove the other) with the well-ordering principle, and yet one is "obviously" true and the other is "obviously" false.

Euclid proved a lot of statements in geometry using several axioms, but the last one was clunky and seemed to be something that ought to be a theorem instead: this held that, given a line and a point not on that line, there was exactly one other line that was parallel to the first line passing through said point. Eventually, though, it was found that there was a reasonable interpretation of geometry where that axiom is not true, whence spherical geometry (parallel lines do not exist) and hyperbolic geometry (many lines can pass through that point and remain parallel).

Another example is in physics: the Crystallographic Restriction Theorem mathematically restricts the kind of forms that crystals could form in. And yet, in the 1980s, several crystals were demonstrated which had five-fold symmetry, which is forbidden by that theorem. The issue is that theorem presupposes that crystals need to be symmetric under linear translations, but there exist forced tilings that have rotational symmetry but not translational symmetry--and these can have five-fold symmetry. (We now call these quasicrystals).

In CS, "unsafe code" amounts to code where programmers did not assert all of the possible preconditions to their code. In contrast to much of mathematics, failing to assert all of the preconditions for safety is remarkably easy in some languages, chiefly C/C++.