To me, the clearest way to think about signed vs. unsigned integers is that different representatives of the integers modulo n are chosen.
For example, for 8-bit signed integers we choose the representatives -128, -127, ..., 127 of the residue classes -128 + 256Z, -127 + 256Z, ..., 127 + 256Z in the ring of integers modulo 256.
For 8-bit unsigned integers, we instead choose the representatives 0, 1, ..., 255.
Mathematically, I do not see how anything is "breaking" as the article claims.
For example, for 8-bit signed integers we choose the representatives -128, -127, ..., 127 of the residue classes -128 + 256Z, -127 + 256Z, ..., 127 + 256Z in the ring of integers modulo 256.
For 8-bit unsigned integers, we instead choose the representatives 0, 1, ..., 255.
Mathematically, I do not see how anything is "breaking" as the article claims.