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A psychological concept. The basic idea is that the human mind can keep track of about seven things at once, or can differentiate between seven or so different (but similar) things.

The phrase comes from the title of a 1956 paper by Harvard professor George A. Miller titled, The Magical Number Seven, Plus or Minus Two: Some Limits on our Capacity for Processing Information, which begins:

My problem is that I have been persecuted by an integer. For seven years this number has followed me around, has intruded in my most private data, and has assaulted me from the pages of our most public journals. This number assumes a variety of disguises, being sometimes a little larger and sometimes a little smaller than usual, but never changing so much as to be unrecognizable. The persistence with which this number plagues me is far more than a random accident. There is, to quote a famous senator, a design behind it, some pattern governing its appearances. Either there really is something unusual about the number or else I am suffering from delusions of persecution.
Miller goes on to present data from a number of cognitive experiments which support the idea (by arriving at the number seven). Topics of the experiments he reviewed included, "span of immediate memory", "capacity for absolute judgements of the position of a dot on a square", and (my favorite) "capacity for absolute judgements of saltiness".