(Just to clarify: this discussion is about the machine learning course. In the future, we should try to keep these as replies in the machine learning discussion thread.)
The curse of dimensionality isn’t meaningful in practice because out space isn’t just a bunch of meaningless cartesian coordinates. We create structure, using trees, neural nets, etc. We regularize using bagging, weight decay, dropout, etc. We find that therefore we actually can add lots of columns without seeing problems in practice.