Q. E. D. is an abbreviation of the Latin phrase "quod erat demonstrandum" (literally, "that which was to have been demonstrated"). This page lists direct English Translations of common Latin phrases, such as Veni vidi vici and et cetera. 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. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language In Logic, an argument is a Set of one or more Declarative sentences (or "propositions") known as the Premises along
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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[1] and Archimedes. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer 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.
Modern 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. The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome.
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. Ethics is a philosophical book written by Baruch Spinoza. It was written in Latin. Baruch or Benedict de Spinoza (ברוך שפינוזה Bento de Espinosa Benedictus de Spinoza ( November 24, 1632 – February 21, Written in Latin, it is considered his magnum opus. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. Magnum opus (sometimes Opus magnum, plural magna opera) from the Latin meaning great work, refers to the best the greatest
The style and system of the book is, as Spinoza says, "demonstrated in geometrical order", with axioms and definitions followed by propositions. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence For Spinoza, this is a considerable improvement over René Descartes's writing style in the Meditations, which follows the form of a diary. Meditations on First Philosophy (subtitled In which the existence of God and the immortality of the soul are demonstrated) is a philosophical treatise written For other uses of the term 'diary' see Diary (disambiguation. [2]
Current usage
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. Tongue-in-cheek is a term used to refer to humor in which a statement or an entire fictional work is not meant to be taken seriously but its lack of seriousness is subtle
Q. E. F.
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). Euclid used this phrase to close propositions which were not precisely "proofs", but rather exemplar constructions. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry 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.
Nonlatin Languages
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". The Ancient Greek language is the historical stage in the development of the Hellenic language family spanning the Archaic (c In modern Greek texts sometimes the "ο. ε. δ. " initials are used at the end of a mathematical proof.
Electronic forms
With computers frequently being used to "write" proofs (see LaTeX), there are several symbolic alternatives in use. LaTeX (ˈleɪtɛ The most popular symbol is ■ (solid black square), also called tombstone or Halmos symbol (after Paul Halmos, who pioneered its use). The tombstone, halmos, or end of proof mark "□" is used in Mathematics to denote the end of a proof in place of the traditional abbreviation Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician 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. In Computing, Unicode is an Industry standard allowing Computers to consistently represent and manipulate text expressed in most of the world's Some authors have adopted variants of this notation with other symbols, such as two forward slashes (//), or simply some vertical white space. In Page layout, Illustration and Sculpture, white space is often referred to as Negative space.
Notes
- ^ Elements 2.5 by Euclid (ed. J. L. Heiberg), retrieved 16 July 2005
- ^ The Chief Works of Benedict De Spinoza, translated by R. Events 622 - The beginning of the Islamic calendar. 1054 - Three Roman legates fractured relations between the Western and Year 2005 ( MMV) was a Common year starting on Saturday (link displays full calendar of the Gregorian calendar. H. M. Elwes, 1951. ISBN 0-486-20250-X.
External links
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