Mathematical Proposition

A mathematical proposition is a mathematical statement whose proof is less complex than that of a mathematical lemma.

• Context:
  • It can be a Proven Mathematical Proposition.
  • Is typically a stepping stone to a more important proof, e.g. of a Mathematical Theorem.
• See: Mathematical Corollary, Semantic Proposition, Proposition.

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References

2013

• http://en.wikipedia.org/wiki/Proposition_%28mathematics%29#Terminology
  • A proposition is a generic term for a theorem of no particular importance. This term sometimes connotes a statement with a simple proof, while the term theorem is usually reserved for the most important results or those with long or difficult proofs. In classical geometry, a proposition may be a construction that satisfies given requirements; for example, Proposition 1 in Book I of Euclid's elements is the construction of an equilateral triangle.

2011

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Proposition
  • … In earlier texts writers have not always made it sufficiently clear whether they are using the term proposition in sense of the words or the "meaning" expressed by the words. To avoid the controversies and ontological implications, the term sentence is often now used instead of proposition to refer to just those strings of symbols that are truthbearers, being either true or false under an interpretation. Strawson advocated the use of the term "statement", and this is the current usage in mathematical logic.

2009

• WordNet.
  • Proposition: (logic) a statement that affirms or denies something and is either true or false

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Proposition_(mathematics)
  • In mathematics, the term proposition is used for a proven statement that is of more than passing interest, but whose proof is neither profound nor difficult.

    The general term for proven mathematical statements is theorem, which also is used in a second, more particular sense, for a proven statement which required some effort, or is in some way a final result. In increasing order of difficulty, the names used for different levels of (general) theorems is approximately: 1. corollary; 2. proposition; 3. lemma; and 4. theorem (particular sense)

    Technically, since a proposition is sometimes followed by a proof, it is a theorem in the general sense, but when the word proposition is used the proof is not challenging enough to call the result a theorem in the particular sense.

• http://en.wiktionary.org/wiki/proposition
  • 1. (uncountable) The act of offering (an idea) for consideration.
  • 2. (countable) An idea or a plan offered.
  • 3. (countable) (in business settings) The terms of a transaction offered.
  • 4. (countable) (logic) The content of an assertion that may be taken as being true or false and is considered abstractly without reference to the linguistic sentence that constitutes the assertion.
  • 5. In some states of the US, a proposed statute or constitutional amendment to be voted on by the electorate.
  • 6. In mathematics, a proposition is an assertion formulated in such a way that it may be proved true or false.

• http://www.uky.edu/~rosdatte/phi120/glossary.htm
  • proposition: A statement. Propositions are sentences or phrases that can be judged to be true or false, for example "The sky is blue" and not "please hurry!"

• http://www.logic-classroom.info/glossary.htm
  • proposition is a form of words in which the predicate is either affirmed or denied of the subject; the meaning expressed by a declarative sentence. (Intro)
• http://planetmath.org/encyclopedia/Lemma.html
  • There is no technical distinction between a lemma, a proposition, and a theorem. A lemma is a proven statement, typically named a lemma to distinguish it as a truth used as a stepping stone to a larger result rather than an important statement in and of itself. Of course, some of the most powerful statements in mathematics are known as lemmas, including Zorn's Lemma, Bezout's Lemma, Gauss' Lemma, Fatou's lemma, etc., so one clearly can't get too much simply by reading into a proposition's name.
  • Even less well-defined is the distinction between a proposition and a theorem. Many authors choose to name results only one or the other, or use both more or less interchangeably. A partially standard set of nomenclature is to use the term proposition to denote a significant result that is still shy of deserving a proper name. In contrast, a theorem under this format would represent a major result, and would often be named in relation to mathematicians who worked on or solved the problem in question.
• http://planetmath.org/encyclopedia/Lemma.html
  • There is no technical distinction between a lemma, a proposition, and a theorem. A lemma is a proven statement, typically named a lemma to distinguish it as a truth used as a stepping stone to a larger result rather than an important statement in and of itself. Of course, some of the most powerful statements in mathematics are known as lemmas, including Zorn's Lemma, Bezout's Lemma, Gauss' Lemma, Fatou's lemma, etc., so one clearly can't get too much simply by reading into a proposition's name.
  • Even less well-defined is the distinction between a proposition and a theorem. Many authors choose to name results only one or the other, or use both more or less interchangeably. A partially standard set of nomenclature is to use the term proposition to denote a significant result that is still shy of deserving a proper name. In contrast, a theorem under this format would represent a major result, and would often be named in relation to mathematicians who worked on or solved the problem in question.

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