Postulate - Revision history

SamHB: And the result is a theorem.

2017-03-03T03:56:36Z

And the result is a theorem.

SamHB: Try to explain (roughly) the difference between an axiom and a postulate.

2017-03-03T03:53:53Z

Try to explain (roughly) the difference between an axiom and a postulate.

Abcqwe: Added much-needed citations and expanded information

2017-03-02T21:20:17Z

Added much-needed citations and expanded information

DavidB4-bot: /* Equivalent statements of the fifth postulate */clean up & uniformity

2016-07-13T17:44:44Z

‎Equivalent statements of the fifth postulate: clean up & uniformity

RSchlafly: /* Euclid's fifth postulate */ negate it to get non-E

2013-06-21T20:06:18Z

‎Euclid's fifth postulate: negate it to get non-E

CSGuy at 02:13, November 14, 2008

2008-11-14T02:13:12Z

NathanG: sp

2008-10-10T21:58:52Z

DanielB: /* The fifth postulate */ made clearer

2008-06-08T01:40:03Z

‎The fifth postulate: made clearer

DanielB: /* The fifth postulate */

2008-06-08T01:39:36Z

‎The fifth postulate

DanielB: /* The fifth postulate */

2008-06-08T01:37:22Z

‎The fifth postulate

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 A '''postulate''' is a statement that is assumed to be true without proof. [[Euclid]], the father of geometry, based ''The Elements'' on ten such statements, divided into five "[[axiom]]s" and five "postulates."<ref>Euclid.  ''The Elements''.</ref>  These are used as foundation for geometry<ref>"1.1:  Definitions, Theorems, and Postulates."  ''Beauty, Rigor, Surprise''.</ref> and occasionally applied to other sciences, such as [[special relativity]].<ref>http://casa.colorado.edu/~ajsh/sr/postulate.html</ref><ref>https://www.nobelprize.org/educational/physics/relativity/postulates-1.html</ref>
-In ordinary discourse about a subject in mathematics or science, the terms "axiom" and "postulate" are often not distinguished&mdash;they are the statements that one takes as true from the outset, that one does not need to prove.  One then uses those to prove other statements.  The difference is that postulates are initial statements relating to the subject matter, such as the geometrical postulates (including Euclid's fifth postulate, below) about angles and lines.  Axioms, on the other hand, tend to be more universal statements about logic, such as "if X=Y, then Y=X".
+In ordinary discourse about a subject in mathematics or science, the terms "axiom" and "postulate" are often not distinguished&mdash;they are the statements that one takes as true from the outset, that one does not need to prove.  One then uses those to prove other statements, which are then called [[theorem]]s.  The difference is that postulates are initial statements relating to the subject matter, such as the geometrical postulates (including Euclid's fifth postulate, below) about angles and lines.  Axioms, on the other hand, tend to be more universal statements about logic, such as "if X=Y, then Y=X".
 ==Euclid's fifth postulate==

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 A '''postulate''' is a statement that is assumed to be true without proof. [[Euclid]], the father of geometry, based ''The Elements'' on ten such statements, divided into five "[[axiom]]s" and five "postulates."<ref>Euclid.  ''The Elements''.</ref>  These are used as foundation for geometry<ref>"1.1:  Definitions, Theorems, and Postulates."  ''Beauty, Rigor, Surprise''.</ref> and occasionally applied to other sciences, such as [[special relativity]].<ref>http://casa.colorado.edu/~ajsh/sr/postulate.html</ref><ref>https://www.nobelprize.org/educational/physics/relativity/postulates-1.html</ref>
 ==Euclid's fifth postulate==
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 Without the fifth postulate many familiar geometric ideas are no longer true, these include,
 * The sum of the interior degrees of a triangle are two right angle (180 degrees)
 [[Category:Mathematics]]
 [[Category:Geometry]]

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-A '''postulate''' is a statement that is assumed to be true without proof. [[Euclid]], the father of geometry, based ''The Elements'' on ten such statements, divided into five "[[axiom]]s" and five "postulates."
+A '''postulate''' is a statement that is assumed to be true without proof. [[Euclid]], the father of geometry, based ''The Elements'' on ten such statements, divided into five "[[axiom]]s" and five "postulates."<ref>Euclid.  ''The Elements''.</ref>  These are used as foundation for geometry<ref>"1.1:  Definitions, Theorems, and Postulates."  ''Beauty, Rigor, Surprise''.</ref> and occasionally applied to other sciences, such as [[special relativity]].<ref>http://casa.colorado.edu/~ajsh/sr/postulate.html</ref><ref>https://www.nobelprize.org/educational/physics/relativity/postulates-1.html</ref>
 ==Euclid's fifth postulate==
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 :Given a line and a point not on that line exactly one line can be drawn through that point which does not intersect the original line.
-That has never seemed as "self-evident" as the others, and in fact for centuries mathematicians thought it could be proved and tried to produce proofs. Eventually, mathematicians realized that this was impossible. If you negated the parallel postulate, you ended up with a perfectly logical, consistent system, a ''non-Euclidean geometry'' that simply happened to describe a kind of geometry ''different'' from plane geometry.
+That has never seemed as "self-evident" as the others, and in fact for centuries mathematicians thought it could be proved and tried to produce proofs. Eventually, mathematicians realized that this was impossible. If you negated the parallel postulate, you ended up with a perfectly logical, consistent system, a ''non-Euclidean geometry'' that simply happened to describe a kind of geometry ''different'' from plane geometry.<ref>"Geometry."  ''The New Encyclopedia Britannica'', 15th ed.</ref><ref>"4.4:  The Playfair Postulate and its Consequences."  Beauty, Rigor, Surprise.</ref>
 ===Equivalent statements of the fifth postulate===

← Older revision		Revision as of 17:44, July 13, 2016
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	[[Category:Mathematics]]		[[Category:Mathematics]]
−	[[~~category~~:Geometry]]	+	[[Category:Geometry]]

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 :Given a line and a point not on that line exactly one line can be drawn through that point which does not intersect the original line.
-That has never seemed as "self-evident" as the others, and in fact for centuries mathematicians thought it could be proved and tried to produce proofs. Eventually, mathematicians realized that this was impossible. If you removed the parallel postulate, you ended up with a perfectly logical, consistent system, a ''non-Euclidean geometry'' that simply happened to describe a kind of geometry ''different'' from plane geometry.
+That has never seemed as "self-evident" as the others, and in fact for centuries mathematicians thought it could be proved and tried to produce proofs. Eventually, mathematicians realized that this was impossible. If you negated the parallel postulate, you ended up with a perfectly logical, consistent system, a ''non-Euclidean geometry'' that simply happened to describe a kind of geometry ''different'' from plane geometry.
 ===Equivalent statements of the fifth postulate===

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-A '''postulate''' is a statement that is assumed to be true without proof. [[Euclid]], the father of geometry, based ''The Elements'' on ten such statements, divided into five "axioms" and five "postulates."
+A '''postulate''' is a statement that is assumed to be true without proof. [[Euclid]], the father of geometry, based ''The Elements'' on ten such statements, divided into five "[[axiom]]s" and five "postulates."
-==The fifth postulate==
+==Euclid's fifth postulate==
 Most of Euclid's axioms and postulates do seem to be "true without needing proof;" for example, "Things which are equal to the same thing are also equal to one another."
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 Euclid's fifth postulate, the ''parallel postulate,'' however, is a little different. It says:
 :It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.
-This is sometimes stated in an equivalent form:
+This is sometimes stated in an equivalent form, known as Playfair's axiom:
 :Given a line and a point not on that line exactly one line can be drawn through that point which does not intersect the original line.