Q.E.D. is an abbreviation of the Latin phrase "quod erat demonstrandum" which means literally, "that which was to be demonstrated". The phrase is written in its abbreviated form at the end of a mathematical proof or philosophical argument, to signify that the last statement deduced was the one to be demonstrated, so the proof is complete.
Etymology and early use
The phrase is a translation into Latin of the original Greek ὅπερ ἔδει δειξαι. (hoper edei deixai
) which was used by many early mathematicians including Euclid
. These mathematicians, in particular Euclid, are credited with founding axiomatic mathematics with its emphasis on establishing truths by logical deduction (rather than experimentation or assertion); their use of this phrase symbolizes this emphasis, as well as marking this important step in the development of mathematical philosophy.
In the European renaissance, mathematical books were typically written in Latin, and phrases such as "quod erat demonstrandum" were often used to conclude proofs.
Perhaps the most famous use of Q.E.D. in a philosophical argument is found in the Ethics of Baruch Spinoza, published posthumously in 1677. Written in Latin, it is considered his magnum opus.
The style and system of the book is, as Spinoza says, "demonstrated in geometrical order", with axioms and definitions followed by propositions. For Spinoza, this is a considerable improvement over René Descartes's writing style in the Meditations, which follows the form of a diary.
Currently, it has become so symbolic of irrefutable logic that "Q.E.D." is occasionally used in non-mathematical contexts as well to intensify assertions; in this context it has little connection with rigorous deduction, however, and is more tongue-in-cheek
It is in this 'tongue-in-cheek' manner that Justice Antonin Scalia of the Supreme Court of the United States, writing the Court's opinion in District of Columbia v. Heller, slip op. at 62 (2008), used the abbreviation. He concluded a paragraph criticizing Justice Steven Breyer's dissenting opinion as follows:
JUSTICE BREYER arrives at his interest-balanced answer: because handgun violence is a problem, because the law is limited to an urban area, and because there were somewhat similar restrictions in the founding period (a false proposition that we have already discussed), the interest-balancing inquiry results in the constitutionality of the handgun ban. QED.
There is another Latin phrase with a slightly different meaning, and less common in usage. Quod erat faciendum
is translated as "which was to have been done." This is usually shortened to Q.E.F.. As with Q.E.D., Q.E.F. is a translation of the Greek geometers' closing ὅπερ ἔδει ποιησαι (hoper edei poiēsai
used this phrase to close propositions which were not precisely "proofs", but rather exemplar constructions. The distinction between Q.E.D. and Q.E.F. is roughly equivalent to the distinction between a proof and an illustration of the proof.
Q.E.D. has acquired many translations; see the language links to the left. In French and German (two of the main languages of Western mathematics) it is respectively C.Q.F.D, for "ce qu'il fallait démontrer" (or sometimes "ce qui finit la démonstration"), and W.Z.B.W, for "was zu beweisen war". There does not appear to be a common formal English equivalent, though the end of a proof may be announced with a simple statement such as "this completes the proof" or a similar locution. The equivalent in Ancient Greek
was "ὅπερ ἔδει δεῖξαι" and could be translated as "which had to be proved". In modern Greek texts sometimes the "ὅ.ἔ.δ." initials are used at the end of a mathematical proof. In Hebrew, the term used is the abbreviation "מ.ש.ל" (pronounced `Mashal`), which stands for "What has needed to be proven".
With computers frequently being used to "write" proofs (see LaTeX
), there are several symbolic alternatives in use. The most popular symbol is ■ (solid black square), also called tombstone
or Halmos symbol
(after Paul Halmos
, who pioneered its use). The tombstone is sometimes open: □ (hollow black square). Unicode
explicitly provides the "End of Proof" character U+220E (∎), but also offers ▮ (U+25AE, black vertical rectangle) and ‣ (U+2023, triangular bullet) as alternatives. Some authors have adopted variants of this notation with other symbols, such as two forward slashes (//), or simply some vertical white space