# Mathematical Theorem

## References

### 2017a

• (Encyclopedia of Mathematics, 2017) ⇒ Retrieved om 2017-05-21 from Theorem. V.E. Plisko (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Theorem&oldid=14225
• A mathematical statement whose truth has been established by means of a proof.

The concept of a theorem developed and became more precise together with the concept of a mathematical proof. With the use of the axiomatic method, the theorems in a theory under consideration are defined as statements deduced in a purely logical way from previously chosen and fixed statements called axioms. Since axioms are assumed to be true, theorems ought to be true as well. A further refinement of the concepts of a proof and a theorem was connected with the investigation, undertaken in mathematical logic, of the concept of a logical consequence, as a result of which for a wide class of mathematical theories it became possible to reduce the process of logical deduction to transformations of formulas, that is, of mathematical statements written in a suitably formalized language, using exactly formulated rules (deduction rules, cf. Derivation rule) about merely the form (and not the content) of the propositions. In formal theories arising in this manner the name proof is given to a finite sequence of formulas each of which either is an axiom or is obtained from certain preceding formulas of this sequence according to the deduction rules. A formula is called a theorem if it is the last formula in a proof.

Such a refinement of the notion of a theorem made it possible to obtain, using rigorous mathematical methods, a series of important results on mathematical theories. In particular, it has been established that axiomatic theories representing substantial chapters of mathematics (for instance, arithmetic) are incomplete, that is, there exist propositions whose truth or falsity cannot be established in a purely logical way on the basis of axioms. As a rule, these theories are undecidable, that is, there is no unique method (algorithm) making it possible to decide whether an arbitrarily given statement is a theorem.

### 2017b

• (Weisstein, 2017) ⇒ Retrieved on 2017-05-21 from: Weisstein, Eric W. "Theorem." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/Theorem.html
• A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof.

Although not absolutely standard, the Greeks distinguished between "problems" (roughly, the construction of various figures) and "theorems" (establishing the properties of said figures; Heath 1956, pp. 252, 262, and 264).

According to the Nobel Prize-winning physicist Richard Feynman (1985), any theorem, no matter how difficult to prove in the first place, is viewed as "trivial" by mathematicians once it has been proven. Therefore, there are exactly two types of mathematical objects: trivial ones, and those which have not yet been proven.

The late mathematician P. Erdős has often been associated with the observation that "a mathematician is a machine for converting coffee into theorems" (e.g., Hoffman 1998, p. 7). However, this characterization appears to be due to his friend, Alfred Rényi (MacTutor, Malkevitch). This thought was developed further by Erdős' friend and Hungarian mathematician Paul Turán, who suggested that weak coffee was suitable "only for lemmas" (MacTutor, Malkevitch).

R. Graham has estimated that upwards of 250000 mathematical theorems are published each year (Hoffman 1998, p. 204).

### 2017c

• (Wikipedia, 2017) ⇒ Retrieved on 2017-05-21 from http://en.wikipedia.org/wiki/Theorem
• In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.[1]

Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.

### 2015A

• (WordNet, 2015) ⇒
• a proposition deducible from basic postulates
• an idea accepted as a demonstrable truth

### 2015B

• en.wiktionary.org/wiki/theorem
• A mathematical statement of some importance that has been proven to be true. Minor theorems are often called propositions. ...

### 2015D

• WordNet.
• a subsidiary proposition that is assumed to be true in order to prove another proposition
• the lower and stouter of the two glumes immediately enclosing the floret in most Gramineae
• the heading that indicates the subject of an annotation or a literary composition or a dictionary entry
• In mathematics, a lemma (plural lemmata or lemmas; from the Greek λήμμα, "lemma" meaning "anything which is received, such as a gift, profit, or a bribe.") is a proven proposition which is used as a stepping stone to a larger result rather than as a statement in-and-of itself.

### 2015E

• en.wikipedia.org/wiki/Lemma_(mathematics)
• In informal logic and argument mapping, a lemma is simultaneously a contention for premises below it and a premise for a contention above it.

### 2015F

• en.wikipedia.org/wiki/Lemma_(logic)
• Farkas' lemma is a result in mathematics used amongst other things in the proof of the Karush-Kuhn-Tucker theorem in nonlinear programming.

### 2015G

• 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.
• The Greek word “lemma” itself means “anything which is received, such as a gift, profit, or a bribe.” According to [1], the plural 'Lemmas' is commonly used. The correct Greek plural of lemma, however, is lemmata. The Greek “Theoria” means “view, or vision" and is clearly linguistically related to the word “theatre.” The apparent relation is that a theorem is a mathematical fact which you see to be true (and can now show others!).
• A somewhat more distinct concept (though still subject to author discretion) is that of a corollary, which is a result that can be considered an immediate consequence of a previous theorem (typically, the preceding theorem in the text).

1. However, both theorems and scientific law are the result of investigations. See Template:Harvnb Introduction, The terminology of Archimedes, p. clxxxii:"theorem (θεὼρνμα) from θεωρεἳν to investigate"