The phrase "two plus two equals five" ("2 + 2 = 5") is a slogan used in George Orwell's Nineteen Eighty-Four[1] as an example of an obviously false dogma one must believe, similar to other obviously false slogans by the Party in the novel. It is contrasted with the phrase "two plus two makes four", the obvious—but politically inexpedient—truth. Orwell's protagonist, Winston Smith, uses the phrase to wonder if the State might declare "two plus two equals five" as a fact; he ponders whether, if everybody believes 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.
Eventually, while undergoing electric torture, Winston declared that he saw five fingers when in fact he only saw four ("Four, five, six — in all honesty I don't know"). The Inner Party interrogator of thought-criminals, O'Brien, says of the mathematically false statement that control over physical reality is unimportant; so long as one controls their own perceptions to what the Party wills, then any corporeal act is possible, in accordance with the principles of doublethink ("Sometimes they are five. Sometimes they are three. Sometimes they are all of them at once").[2]
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George Orwell had used the concept before publishing Nineteen Eighty-Four. During his career at the BBC, he became familiar with the methods of Nazi propaganda. In his essay "Looking Back on the Spanish War",[3] published in 1943 (six years before the publication of Nineteen Eighty-Four,) Orwell wrote:
| “ | Nazi theory indeed specifically denies that such a thing as "the truth" exists. […] The implied objective of this line of thought is a nightmare world in which the Leader, or some ruling clique, controls not only the future but the past. If the Leader says of such and such an event, "It never happened"—well, it never happened. If he says that two and two are five—well, two and two are five. This prospect frightens me much more than bombs […][3] | ” |
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 debatably hyperbolic display of loyalty to Adolf Hitler, declared, "If the Führer wants it, two and two makes five!"[4] 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?[5] | ” |
In Fyodor Dostoyevsky's Notes from Underground, the protagonist implicitly supports the idea of two times two making five, spending several paragraphs considering the implications of rejecting the statement "two times two makes four."
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."
Dostoyevsky 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.[6]
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."
Victor Hugo here is echoing earlier French thought — Sieyes, in his "What is the Third Estate?" uses the phrase, "Consequently if it be claimed that under the French constitution, 200,000 individuals out of 26 million citizens constitute two-thirds of the common will, only one comment is possible: it is a claim that two and two make five."
It is very plausible that Dostoyevsky 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 changed his opinions to something that fit no 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.[7]
If "two plus two equals five" is used as a propositional statement about integers, it must be either true or false.[8]
However, by redefining what we mean by "2" and "5", we can make the answer anything we like. For instance, if by "2" we mean "any real number which rounds to 2" and by "5" we mean "any real number which rounds to 5", then from 2.4+2.4=4.8 we conclude that "2+2=5". A difficulty with this approach, of course, is that it runs afoul of Lincoln's dictum that calling a tail a leg doesn't make it one. The question at this point gets bogged down in the philosophy of language, and in any event has nothing to do with mathematics.
In his play Dom Juan, Moliere's title character is asked what he believes. He answers that he believes that two plus two equals four.[9] Belief is the psychological state in which an individual holds a proposition or premise to be true.[10] A belief is separate from knowledge.[11][12] Were certain absolute knowledge to exist, belief in an existential claim would be unnecessary. Moliere seeks the freedom to believe that two plus two equals four. Orwell seeks the freedom to say that two plus two equals four, as an objective fact which the Party cannot touch.
Rene Descartes' realm of pure ideas considers that self-evident ideas such as two plus two equals four may in fact have no reality outside the mind. According to the first meditation, the standard of truth is self-evidence of clear and distinct ideas. However, Descartes questions the correspondence of these ideas to reality.[13]
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