The most prolific contemporary proponent of this view is probably Norman Wildberger, who takes a very unconventional approach and tries to build everything with

The most prolific contemporary proponent of this view is probably Norman Wildberger, who takes a very unconventional approach and tries to build everything without real numbers, which he doesn't believe make physical or logical sense and therefore shouldn't be used as a basis for anything. He's very nearly a minority of one, mind you.