Comment by lupusreal

I think the most coherent answer is that it was simply a throwback to 20th century science fiction, in which psychic powers were commonly treated as "real in the future". The Matrix in particular borrowed a lot from anime and eastern mysticism, so a break from strict materialism isn't too out of place. It's just part of the style of this kind of media.

(Psychics in sci-fi: Foundation, Ringworld, Akira and about a million other animes, The Demolished Man, The Stars My Destination, Dune, loads of Phillip K Dick, Starship Troopers,... If you read a lot of 20th century sci-fi it comes up A LOT.)