Orwell's protagonist, Winston Smith, uses the phrase to wonder if the State might declare "two plus two makes five" as a fact; he ponders that, if everybody believes in it, does that make it true? Smith writes, "Freedom is the freedom to say that two plus two make four. If that is granted, all else follows." Later in the novel, Smith attempts to use doublethink to teach himself that the statement "2 + 2 = 5" is true, or at least as true as any other answer one could come up with.
Later in the novel, Winston Smith is electroshocked into declaring that he saw five fingers when in fact he only saw four ("Four! Five! Four! I don't know!").
In the view of most of Orwell's biographers, the main source for this was Assignment in Utopia by Eugene Lyons, an account of his time in the Soviet Union. This contains a chapter "Two Plus Two Equals Five", which was a slogan used by Stalin's government to predict that the Five year plan would be completed in four years, which for a time appeared widely in Moscow.
However, Orwell may also have been influenced by Nazi Reichsmarschall Hermann Göring, who once, in a debatable hyperbolic display of loyalty to Adolf Hitler, declared, "If the Führer wants it, two and two makes five! In Nineteen Eighty-Four, Orwell writes:
In the end the Party would announce that two and two made five, and you would have to believe it. It was inevitable that they should make that claim sooner or later: the logic of their position demanded it. Not merely the validity of experience, but the very existence of external reality, was tacitly denied by their philosophy. The heresy of heresies was common sense. And what was terrifying was not that they would kill you for thinking otherwise, but that they might be right. For, after all, how do we know that two and two make four? Or that the force of gravity works? Or that the past is unchangeable? If both the past and the external world exist only in the mind, and if the mind itself is controllable—what then?
Yevgeny Zamyatin's novel We, an acknowledged precursor to Nineteen Eighty-Four, has an advocate of the totalitarian state discussing the sum "two times two makes four" in an argument against freedom. The character argues that it would "be an absurdity if these two and two were to get notions about some sort of freedom - i.e., about that which is clearly an error".
His purpose is not ideological, however. Instead, he proposes that it is the free will to choose or reject the logical as well as the illogical that makes mankind human. He adds: "I admit that two times two makes four is an excellent thing, but if we are to give everything its due, two times two makes five is sometimes a very charming thing too."
Dostoevsky was writing in 1864. However, according to Roderick T. Long, Victor Hugo had used the phrase back in 1852. He objected to the way in which the vast majority of French voters had backed Napoleon III, endorsing the way liberal values had been ignored in Napoleon III's coup.
Victor Hugo said "Now, get seven million five hundred thousand votes to declare that two and two make five, that the straight line is the longest road, that the whole is less than its part; get it declared by eight millions, by ten millions, by a hundred millions of votes, you will not have advanced a step."
It's very plausible that Dostoevsky had this in mind. He had been sentenced to death for his participation in a radical intellectual discussion group. The sentence was commuted to imprisonment in Siberia, and he then changed his opinions to something that doesn't fit any conventional labels.
The idea seems to have been significant to Russian literature and culture. Ivan Turgenev wrote in prayer, one of his Poems in Prose "Whatever a man prays for, he prays for a miracle. Every prayer reduces itself to this: Great God, grant that twice two be not four." Also similar sentiments are said to be among Leo Tolstoy's last words when urged to convert back to the Russian Orthodox Church: "Even in the valley of the shadow of death, two and two do not make six." Even turn-of-the-century Russian newspaper columnists used the phrase to suggest the moral confusion of the age (e.g. Novoe vremia (New Times), 31 October 1900.
In measurement in science, the number of significant digits is usually encoded in the way a number is written. That is, under certain conditions, "2" has only one significant digit, which means it represents a measurement with a margin of error of 1, which means the actual value may lie between 1.0... and 3.0. When adding such measurements together, the margins of error are also added in quadrature, so (2±1) + (2±1) = 4 ± (1.0^2 + 1.0^2)^1/2 = 4 ± 1. And since "5" actually means 5 ± 1, these margins clearly overlap and one could jokingly argue that the numbers are the same. A popular phrasing of this statement is "2 + 2 = 5 for large values of 2 or small values of 5". When adding more precise measurements, for example 2.0 + 2.0, the margin of error is smaller and the maximum number that could be "reached" would in this case be 4.1. See also approximation and floating point.
Similar concepts are explored in the following songs:
In the 1980's the administration of the Ben-Gurion University, at Beersheba, Israel, enforced a policy of a complete ban on any demonstrations or political activities on campus - offenders being punished by suspension from studies or in some cases permanent expulsion. Dissident philosophy students sought to challenge the ban by holding a demonstration with signs reading "2 + 2 = 4". The humourless university administration applied to them the usual penalties