Q.E.D. or QED is an initialism of the Latin phrasequod erat demonstrandum, meaning "that which was to be demonstrated". Literally it states "what was to be shown".[1] Traditionally, the abbreviation is placed at the end of mathematical proofs and philosophicalarguments in print publications, to indicate that the proof or the argument is complete.
The phrase quod erat demonstrandum is a translation into Latin from the Greekὅπερ ἔδει δεῖξαι (hoper edei deixai; abbreviated as ΟΕΔ). Translating from the Latin phrase into English yields "that was to be demonstrated". However, translating the Greek phrase ὅπερ ἔδει δεῖξαι can produce a slightly different meaning. In particular, since the verb "δείκνυμι" also means to show or to prove,[2] a different translation from the Greek phrase would read "The very thing it was required to have shown."[3]
The Greek phrase was used by many early Greek mathematicians, including Euclid[4] and Archimedes.
Philippe van Lansberge's 1604 Triangulorum Geometriæ used quod erat demonstrandum to conclude some proofs; others ended with phrases such as sigillatim deinceps demonstrabitur, magnitudo demonstranda est, and other variants.
During the European Renaissance, scholars often wrote in Latin, and phrases such as Q.E.D. were often used to conclude proofs.
Spinoza's original text of Ethics, Part 1, Q.E.D. is used at the end of Demonstratio of Propositio III on the right hand page
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.[11] Written in Latin, it is considered by many to be Spinoza's magnum opus. The style and system of the book are, 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.[12]
Difference from Q.E.F.
There is another Latin phrase with a slightly different meaning, usually shortened similarly, but being less common in use. Quod erat faciendum, originating from the Greek geometers' closing ὅπερ ἔδει ποιῆσαι (hoper edei poiēsai), meaning "which had to be done".[13] Because of the difference in meaning, the two phrases should not be confused.
Euclid used the Greek original of Quod Erat Faciendum (Q.E.F.) to close propositions that were not proofs of theorems, but constructions of geometric objects.[14] For example, Euclid's first proposition showing how to construct an equilateral triangle, given one side, is concluded this way.[15]
Equivalent forms
There is no common formal English equivalent, although the end of a proof may be announced with a simple statement such as "thus it is proved", "this completes the proof", "as required", "as desired", "as expected", "hence proved", "ergo", "so correct", or other similar locutions.
Due to the paramount importance of proofs in mathematics, mathematicians since the time of Euclid have developed conventions to demarcate the beginning and end of proofs. In printed English language texts, the formal statements of theorems, lemmas, and propositions are set in italics by tradition. The beginning of a proof usually follows immediately thereafter, and is indicated by the word "proof" in boldface or italics. On the other hand, several symbolic conventions exist to indicate the end of a proof.
While some authors still use the classical abbreviation, Q.E.D., it is relatively uncommon in modern mathematical texts. Paul Halmos claims to have pioneered the use of a solid black square (or rectangle) at the end of a proof as a Q.E.D. symbol,[16] a practice which has become standard, although not universal. Halmos noted that he adopted this use of a symbol from magazine typography customs in which simple geometric shapes had been used to indicate the end of an article, so-called end marks.[17][18] This symbol was later called the tombstone, the Halmos symbol, or even a halmos by mathematicians. Often the Halmos symbol is drawn on chalkboard to signal the end of a proof during a lecture, although this practice is not so common as its use in printed text.
The tombstone symbol appears in TeX as the character (filled square, \blacksquare) and sometimes, as a (hollow square, \square or \Box).[19] In the AMS Theorem Environment for LaTeX, the hollow square is the default end-of-proof symbol. Unicode explicitly provides the "end of proof" character, U+220E (∎). Some authors use other Unicode symbols to note the end of a proof, including, ▮ (U+25AE, a black vertical rectangle), and ‣ (U+2023, a triangular bullet). Other authors have adopted two forward slashes (//, ) or four forward slashes (////, ).[20] In other cases, authors have elected to segregate proofs typographically—by displaying them as indented blocks.[21]
Modern humorous use
In Joseph Heller's 1961 book Catch-22, the Chaplain, having been told to examine a forged letter allegedly signed by him (which he knew he didn't sign), verified that his name was in fact there. His investigator replied, "Then you wrote it. Q.E.D." The chaplain said he did not write it and that it was not his handwriting, to which the investigator replied, "Then you signed your name in somebody else's handwriting again."[22]
In the 1978 science-fiction radio comedy, and later in the television, novel, and film adaptations of The Hitchhiker's Guide to the Galaxy, "Q.E.D." is referred to in the Guide's entry for the babel fish, when it is claimed that the babel fish – which serves the "mind-bogglingly" useful purpose of being able to translate any spoken language when inserted into a person's ear – is used as evidence for existence and non-existence of God. The exchange from the novel is as follows: "'I refuse to prove I exist,' says God, 'for proof denies faith, and without faith I am nothing.' 'But,' says Man, 'The babel fish is a dead giveaway, isn't it? It could not have evolved by chance. It proves you exist, and so therefore, by your own arguments, you don't. QED.' 'Oh dear,' says God, 'I hadn't thought of that,' and promptly vanishes in a puff of logic."[23]
In Neal Stephenson's 1999 novel Cryptonomicon, Q.E.D. is used as a punchline to several humorous anecdotes, in which characters go to great lengths to prove something non-mathematical.[24]
Singer-songwriter Thomas Dolby's 1988 song "Airhead" includes the lyric, "Quod erat demonstrandum, baby," referring to the self-evident vacuousness of the eponymous subject; and in response, a female voice delightedly squeals, "Oooh... you speak French!" [25]
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.
Euclid was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the Elements treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved new innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics.
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.
QED may refer to:
In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid.
In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics.
A mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.
The QED manifesto was a proposal for a computer-based database of all mathematical knowledge, strictly formalized and with all proofs having been checked automatically.
The Elements is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid c. 300 BC. It is a collection of definitions, postulates, propositions, and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science, and its logical rigor was not surpassed until the 19th century.
Quod may refer to:
Paul Richard Halmos was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator theory, ergodic theory, and functional analysis. He was also recognized as a great mathematical expositor. He has been described as one of The Martians.
In mathematics, the tombstone, halmos, end-of-proof, or Q.E.D. symbol "∎" is a symbol used to denote the end of a proof, in place of the traditional abbreviation "Q.E.D." for the Latin phrase "quod erat demonstrandum". It is inspired by the typographic practice of end marks, an element that marks the end of an article.
In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers, namely:
The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this is common English, but with a specific non-obvious meaning when used in a mathematical sense.
The following table lists many specialized symbols commonly used in modern mathematics, ordered by their introduction date. The table can also be ordered alphabetically by clicking on the relevant header title.
Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.
In logic, reductio ad absurdum, also known as argumentum ad absurdum or apagogical arguments, is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction.
Q.E.D. is the second widely available studio album by American blues rock musician and computer scientist Jim Allchin. It was released on September 10, 2013 by Sandy Key Music. The title of the album is a reference to quod erat demonstrandum or Q.E.D which is sometimes used to denote the completion of a mathematical proof.
↑ This (generally accepted) claim was made in Halmos's autobiography, I Want to Be a Mathematician. The first usage of the solid black rectangle as an end-of-proof symbol appears to be in Halmos's Measure Theory (1950). The intended meaning of the symbol is explicitly given in the preface.
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