In "COMPUTING MACHINERY AND INTELLIGENCE" Alan Turing says the following:
I assume that the reader is familiar with the idea of extrasensory perception, and the meaning of the four items of it, viz., telepathy, clairvoyance, precognition and psychokinesis. These disturbing phenomena seem to deny all our usual scientific ideas. How we should like to discredit them! Unfortunately the statistical evidence, at least for telepathy, is overwhelming. It is very difficult to rearrange one's ideas so as to fit these new facts in. Once one has accepted them it does not seem a very big step to believe in ghosts and bogies. The idea that our bodies move simply according to the known laws of physics, together with some others not yet discovered but somewhat similar, would be one of the first to go.
It seems like he believed that telepathy was a scientific fact, or close to it. Obviously this is not a widely accepted idea any more. Was it popular at the time? What led to this?
A link to the whole paper below: http://www.loebner.net/Prizef/TuringArticle.html
In the Victorian and early Edwardian period (and actually, depending on who you ask, through the 1970s) there was a lot of serious research into the phenomena Turing refers to in that passage. For instance, the Society for Psychical Research applied the scientific method to psychics, mediums, and so on, and published hundreds of case studies of "real" psychics, telepaths, clairvoyants, and the like (the collection is a brick or two, bound up in hardcovers and nominally on my comps list).
I believe there's even a relatively famous passage in some of Carl Jung's work in which he refers to psychokinesis, which he had to shrug at and say "I dunno man" while Freud, who was also in the room at the time, disavowed.
My favorite anecdotal bit is that Algernon Blackwood, a famous horror writer of the early 20th century, was actually hired by the Psychical Research Society to investigate haunted houses -- ghost hunters 1910s style -- just because he was so good at portraying them in fiction.
The thing about such research is that, like anything else, it's almost impossible to actually prove a negative. You can't actually prove telepathy doesn't exist, just that subject A or subject B does not exhibit such abilities. So it makes sense for it to be a legitimate avenue of scientific inquiry, particularly in such a historical period, recovering from some of the larger societal blows to traditional spirituality suffered.