Telepathy

Are you thinking what we're thinking? Telepathy is the alleged transfer of information between individuals' minds without any apparent detectable connection (physical, electronic, electromagnetic). Despite considerable testing, "telepathic" individuals have been unable to show true telepathy.

Several governments have studied telepathy with the intention of either using it as a secure means of communication or as a method of obtaining information by "reading minds".

Telepathic mind-reading has often been performed as a stage act. Many telepaths simply use suggestive questions to put ideas in the mind of their subjects (as demonstrated by Derren Brown in numerous performances).

Science fiction stories have used telepathy as a plot device countless times, but quite often it is electronically aided. The modern cell phone comes pretty close to doing the job.

Research into brain function, using either introduced probes or external detectors, to control electronic devices is seen by some as foreshadowing telepathy. But of course if we achieve it through technology, the woo crowd will lose interest in it because then it'll actually work it won't be special, mysterious, or mystical any more.

The inverse square law
If telepathy exists and is based on any ordinary physical force it should follow the inverse square law. Any real physical force transmitting telepathic information should weaken progressively as the distance between the communicating subjects increases. The inverse square law is based on logic and geometry. Real telepathy involving minds at a distance would need complex explanations involving for example higher dimensions than the familiar three. Occam's razor suggests that is unlikely. On the other hand apparent telepathy could be based on publication bias: results showing apparent telepathy are more likely to be published. Apparent, unreal telepathy between minds far from each other is just as likely as apparent telepathy involving minds close together. Distance of subjects is unlikely to affect publication bias.

Simple explanations
In some cases, people may well think the same thing, due to the same external influences. For example, if one takes a busload of people on a trip through the countryside, it is likely the individuals will share thoughts about fields, farms, livestock etc. later in the day. This accounts for twins' "telepathy", as they are often raised in the same environment, with the same influences and many shared experiences. If you shared most of your waking hours with the same person, who also had some or all of your DNA, you'd probably get pretty good at guessing what they're thinking and feeling.

Confirmation bias also accounts for why people remember apparent "telepathy" and coincidence, but not the numerous examples to the contrary. Publication bias can also lead to cases of apparent telepathy being published while negative results remain unpublished.