So I was reading this article because I'm new to this stuff and don't quite understand the ways in which quantum mechanics and GR contradict each other. Why does general relativity require space to be continuous?
From your link:
General relativity needs space to be “smooth”, or at the very least continuous. So if you have two points side by side, then no matter how close you bring them together you can still tell which one is on the right or left.
True: Continuous space time variables are necessary for Lorenz transformations to work, and the validity of Lorenz transformations has been demonstrated innumerable times. They are also a part of General Relativity for flat spaces.
Quantum mechanically you have to deal with position uncertainty. At very small scales you can’t tell which is right or left.
Not true :You can give a probability for all variables under measurement, including left and right. In an accumulation of measurements you can tell right from left.
In addition (as the name implies) QM requires everything to be “quantized”, or show up in discrete pieces.
Not true: Quantization appears under specific solutions of boundary condition problems. Space and time are continuous for quantum mechanics which is compatible with the Lorenz transformations.
In fact already there exist theories which quantize General Relativity and are proposed for particle physics experiments, string theories.