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pos·tu·late • v. / ˈpäschəˌlāt/ [tr.] 1. suggest or assume the existence, fact, or truth of (something) as a basis for reasoning, discussion, or belief: his theory postulated a rotatory movement for hurricanes | he postulated that the environmentalists might have a case. 2. (in ecclesiastical law) nominate or elect (someone) to an ecclesiastical office subject to the sanction of a higher authority. • n. / ˈpäschələt/ formal a thing suggested or assumed as true as the basis for reasoning, discussion, or belief: perhaps the postulate of Babylonian influence on Greek astronomy is incorrect. ∎  Math. an assumption used as a basis for mathematical reasoning. DERIVATIVES: pos·tu·la·tion / ˌpäschəˈlāshən/ n.

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postulate

postulate † demand XVI; proposition claimed to be granted, (geom.) problem of self-evident nature XVII. — L. postulātum (also used), sb. use of n. pp. of postulāre, prob. f. base of poscere (see PRAY).
So vb. XVI.

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postulate

postulate: see axiom.

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postulate

postulate •circumvallate • bedplate • template •breastplate • nameplate • faceplate •chelate • fishplate • sibilate • jubilate •flagellate • legislate • invigilate •assimilate, dissimilate •depilate, epilate •fibrillate •correlate, intercorrelate •vacillate • tessellate • oscillate •cantillate •hyperventilate, ventilate •titillate • scintillate • constellate •mutilate • oblate • hotplate •electroplate • bookplate • footplate •congratulate •confabulate, tabulate •ambulate, circumambulate, perambulate •adulate • coagulate •strangulate, triangulate •ejaculate •calculate, miscalculate •emasculate • granulate • encapsulate •regulate • speculate • emulate •infibulate • acidulate •articulate, gesticulate, matriculate •simulate, stimulate •manipulate, stipulate •insulate • capitulate •discombobulate • modulate •flocculate, inoculate •osculate •copulate, populate •expostulate, postulate •ovulate • formulate • ululate •accumulate, cumulate •undulate • pustulate • circulate •lanceolate •annihilate, violate •number plate • fingerplate • escalate •percolate • immolate •crenellate (US crenelate) •extrapolate • copperplate •interpellate, interpolate •desolate • insufflate • isolate •apostolate • contemplate •braggart, faggot (US fagot), maggot •legate •bigot, gigot, Piggott, spigot •ingot • profligate • aggregate • yogurt •conjugate • abrogate • surrogate •ergot, virgate •Bagehot • patriarchate • wainscot •Sickert • predicate • syndicate •certificate, pontificate •Calicut • delicate • silicate • triplicate •duplicate, quadruplicate •intricate • Connecticut • Alcott •ducat • advocate •ballot, palate •charlotte, harlot •appellate, Helot, prelate, zealot •flagellate • distillate •Pilate, pilot •copilot • gyropilot • autopilot •triangulate •ejaculate, immaculate •amulet • spatulate •articulate, denticulate •consulate, proconsulate •postulate • ungulate •inviolate, ultraviolet •chocolate • cardinalate • desolate •isolate • disconsolate • Merlot

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Postulate

POSTULATE

A postulate is an assumption advanced with the claim that it be taken for granted as axiomatic. In reference to further investigation it is a statement so assumed as to require no proof of its validity. In the development of theory, it corresponds to the first principles of philosophy. Practically, a postulate is roughly the same as a hypothesis except that the postulate is considered to be the idea content of the assumption and the hypothesis the logical statement of the postulate. In scientific theory postulates are generally either laws or principles that are considered as established, e.g., Newton's laws of motion, or convenient concepts that can neither be proved nor disproved, e.g., the principle of the conservation of energy.

See Also: axiomatic system.

[l. a. foley]

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Postulate

Postulate

Resources

A postulate is an assumption, that is, a proposition or statement that is assumed to be true without any proof. Postulates are the fundamental propositions used to prove other statements known as theorems. Once a theorem has been proven it is may be used in the proof of other theorems. In this way, an entire branch of mathematics can be built up from a few postulates. Postulate is synonymous with axiom, although sometimes axiom is taken to mean an assumption that applies to all branches of mathematics; in that case a postulate is taken to be an assumption specific to a given theory or branch of mathematics. Euclidean geometry provides a classic example: Euclid based his geometry on five postulates and five common notions, of which the postulates are assumptions specific to geometry, and the common notions are completely general axioms.

The five postulates of Euclid that pertain to geometry are specific assumptions about lines, angles, and other geometric concepts. They are:

  1. Any two points describe a line.
  2. A line is infinitely long.
  3. A circle is uniquely defined by its center and a point on its circumference.
  4. Right angles are all equal.
  5. Given a point and a line not containing the point, there is one and only one parallel to the line through the point.

The five common notions of Euclid have application in every branch of mathematics; they are:

  1. Two things that are equal to a third are equal to each other.
  2. Equal things having equal things added to them remain equal.
  3. Equal things having equal things subtracted from them have equal remainders.
  4. Any two things that can be shown to coincide with each other are equal.
  5. The whole is greater than any part.

On the basis of these 10 assumptions, Euclid produced the Elements, a 13-volume treatise on geometry (published c. 300 BC) containing some 400 theorems, now referred to collectively as Euclidean geometry.

When developing a mathematical system through logical deductive reasoning any number of postulates may be assumed. Sometimes in the course of proving theorems based on these postulates a theorem turns out to be the equivalent of one of the postulates. Thus, mathematicians usually seek the minimum number of postulates on which to base their reasoning. It is interesting to note that for centuries following publication of the Elements, mathematicians believed that Euclids fifth postulate, sometimes called the parallel postulate, could logically be deduced from the first four. Not until the nineteenth century did mathematicians recognize that the five postulates did indeed result in a logically consistent geometry, and that replacement of the fifth postulate with different assumptions led to other consistent geometries.

Postulates figure prominently in the work of the Italian mathematician Guiseppe Peano (1858-1932), who formalized the language of arithmetic by choosing three basic concepts: zero; number (meaning the non-negative integers); and the relationship is the successor of. In addition, Peano assumed that the three concepts obeyed the five following axioms or postulates:

  1. Zero is a number.
  2. If b is a number, the successor of b is a number.
  3. Zero is not the successor of a number.
  4. Two numbers of which the successors are equal are themselves equal.
  5. If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.

Based on these five postulates, Peano was able to derive five fundamental laws of arithmetic, the Peano axioms, that provided not only a formal foundation for arithmetic but for many of the constructions upon which algebra depends.

Indeed, during the nineteenth century, virtually every branch of mathematics was reduced to a set of postulates and resynthesized in logical deductive fashion. The result was to change the way mathematics is viewed. Prior to the nineteenth century mathematics had been seen solely as a means of describing the physical universe. By the end of the century, however, mathematics came to be viewed more as a means of deriving the logical consequences of a collections of axioms.

In the twentieth century, a number of important discoveries in mathematics and logic showed the limitations of proof from postulates, thereby invalidating Peanos axioms. The best known of these is Gödels theorem, formulated in the 1930s by the Austrian mathematician Kurt Gödel (1906-1978). Gödel demonstrated that if a system contained Peanos postulates, or an equivalent, the system was either inconsistent (a statement and its opposite could be proved) or incomplete (there are true statements that cannot be derived from the postulates).

See also Logic, symbolic.

Resources

BOOKS

Boyer, Carl B. A History of Mathematics. 2nd ed. Revised by Uta C. Merzbach. New York: Wiley, 1991.

Paulos, John Allen. Beyond Numeracy, Ruminations of a Numbers Man. New York: Knopf, 1991.

Smith, Stanley A., Charles W. Nelson, Roberta K. Koss, Mervin L. Keedy, and Marvin L. Bittinger. Addison Wesley Informal Geometry. Reading MA: Addison Wesley, 1992.

BOOKS

Elements. Wolfram MathWorld <http://mathworld.wolfram.com/Elements.html> [October 9, 2006).

J. R. Maddocks

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Postulate

Postulate

A postulate is an assumption, that is, a proposition or statement, that is assumed to be true without any proof . Postulates are the fundamental propositions used to prove other statements known as theorems. Once a theorem has been proven it is may be used in the proof of other theorems. In this way, an entire branch of mathematics can be built up from a few postulates. Postulate is synonymous with axiom, though sometimes axiom is taken to mean an assumption that applies to all branches of mathematics, in which case a postulate is taken to be an assumption specific to a given theory or branch of mathematics. Euclidean geometry provides a classic example. Euclid based his geometry on five postulates and five "common notions," of which the postulates are assumptions specific to geometry, and the "common no tions" are completely general axioms.

The five postulates of Euclid that pertain to geometry are specific assumptions about lines, angles, and other geometric concepts. They are:

  1. Any two points describe a line.
  2. A line is infinitely long.
  3. A circle is uniquely defined by its center and a point on its circumference.
  4. Right angles are all equal.
  5. Given a point and a line not containing the point, there is one and only one parallel to the line through the point.

The five "common notions" of Euclid have application in every branch of mathematics, they are:

  1. Two things that are equal to a third are equal to each other.
  2. Equal things having equal things added to them remain equal.
  3. Equal things having equal things subtracted from them have equal remainders.
  4. Any two things that can be shown to coincide with each other are equal.
  5. The whole is greater than any part.

On the basis of these ten assumptions, Euclid produced the Elements, a 13 volume treatise on geometry (published c. 300 b.c.) containing some 400 theorems, now referred to collectively as Euclidean geometry.

When developing a mathematical system through logical deductive reasoning any number of postulates may be assumed. Sometimes in the course of proving theorems based on these postulates a theorem turns out to be the equivalent of one of the postulates. Thus, mathematicians usually seek the minimum number of postulates on which to base their reasoning. It is interesting to note that, for centuries following publication of the Elements, mathematicians believed that Euclid's fifth postulate, sometimes called the parallel postulate, could logically be deduced from the first four. Not until the nineteenth century did mathematicians recognize that the five postulates did indeed result in a logically consistent geometry, and that replacement of the fifth postulate with different assumptions led to other consistent geometries.

Postulates figure prominently in the work of the Italian mathematician Guiseppe Peano (1858-1932), formalized the language of arithmetic by choosing three basic concepts: zero ; number (meaning the non-negative integers ); and the relationship "is the successor of." In addition, Peano assumed that the three concepts obeyed the five following axioms or postulates:

  1. Zero is a number.
  2. If b is a number, the successor of b is a number.
  3. Zero is not the successor of a number.
  4. Two numbers of which the successors are equal are themselves equal.
  5. If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.

Based on these five postulates, Peano was able to derive the fundamental laws of arithmetic. Known as the Peano axioms, these five postulates provided not only a formal foundation for arithmetic but for many of the constructions upon which algebra depends.

Indeed, during the nineteenth century, virtually every branch of mathematics was reduced to a set of postulates and resynthesized in logical deductive fashion. The result was to change the way mathematics is viewed. Prior to the nineteenth century mathematics had been seen solely as a means of describing the physical universe. By the end of the century, however, mathematics came to be viewed more as a means of deriving the logical consequences of a collections of axioms.

In the twentieth century, a number of important discoveries in the fields of mathematics and logic showed the limitation of proof from postulates, thereby invalidating Peano's axioms. The best known of these is Gödel's theorem, formulated in the 1930s by the Austrian mathematician Kurt Gödel (1906-1978). Gödel demonstrated that if a system contained Peano's postulates, or an equivalent, the system was either inconsistent (a statement and its opposite could be proved) or incomplete (there are true statements that cannot be derived from the postulates).

See also Logic, symbolic.


Resources

books

Boyer, Carl B. A History of Mathematics. 2nd ed. Revised by Uta C. Merzbach. New York: Wiley, 1991.

Paulos, John Allen. Beyond Numeracy, Ruminations of a Numbers Man. New York: Knopf, 1991.

Smith, Stanley A., Charles W. Nelson, Roberta K. Koss, Mervin L. Keedy, and Marvin L. Bittinger. Addison Wesley Informal Geometry. Reading MA: Addison Wesley, 1992.


J. R. Maddocks

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