Theorem - Mathematics and Logic

In mathematics and logic, a proposition or statement that is demonstrated. In geometry, a proposition is commonly considered as a problem (a construction to be effected) or a theorem (a statement to be proved). The statement “If two lines intersect, each pair of vertical angles is equal,” for example, is a theorem. The so-called fundamental theorem of algebra asserts that every (complex) polynomial equation in one variable has at least one complex root or solution. The Greeks also recognized a proposition lying between a theorem and a problem, the porism, directed to producing or finding what is proposed.

In formal settings, an necessary property of theorems is that they are derivable using a fixed set of supposition rules and axioms without any additional assumptions. This is not a matter of the semantics of the language: the expression that results from a source is a syntactic consequence of all the expressions that precede it. In mathematics, the derivation of a theorem is often interpreted as a proof of the fact of the resulting expression, but dissimilar deductive systems can yield other interpretations, depending on the meanings of the derivation rules.

The proofs of theorems have two components, called the hypotheses and the conclusions. The proof of a mathematical theorem is a logical argument signifying that the conclusions are a necessary outcome of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions. The concept of a theorem is therefore fundamentally deductive, in contrast to the concept of a scientific theory, which is empirical.

Formal and informal notions

Logically most theorems are of the form of an indicative conditional: if A, then B. Such a theorem does not state that B is always true, only that B must be true if A is true. In this case A is called the hypothesis of the theorem (note that "hypothesis" here is something very different from a conjecture) and B the conclusion. The theorem "If n is an even natural number then n/2 is a natural number" is a typical example in which the hypothesis is that n is an even natural number and the conclusion is that n/2 is also a natural number.

In order to be proven, a theorem must be expressible as a precise, formal statement. Nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader will be able to produce a formal statement from the informal one. In addition, there are often hypotheses which are implicit in context, rather than explicitly stated.

It is common in mathematics to choose a number of hypotheses that are assumed to be true within a given theory, and then proclaim that the theory consists of all theorems provable using those hypotheses as assumption. In this case the hypotheses that form the foundational basis are called the axioms (or postulates) of the theory. The field of mathematics known as proof theory studies formal axiom systems and the proofs that can be performed within them.

Relation with scientific theories

Theorems in mathematics and theories in science are basically different in their epistemology. A scientific theory cannot be proven; its key attribute is that it is falsifiable, that is, it makes predictions about the natural world that are testable by experiments. Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its exactness or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories.

Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing the proof. For example, the Collatz conjecture has been verified for start values up to about 2.88 × 1018. The Riemann hypothesis has been verified for the first 10 trillion zero's of the zeta function. Neither of these statements is considered to be proven.

Such evidence does not constitute proof. For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e., a natural number n for which the Mertens function M(n) equals or exceeds the square root of n) is known: all numbers less than 1014 have the Mertens property, and the smallest number which does not have this property is only known to be less than the exponential of 1.59 × 1040, which is approximately 10 to the power 4.3 × 1039. Since the number of particles in the universe is generally considered to be less than 10 to the power 100 (a googol), there is no hope to find an explicit counterexample by exhaustive search at present.

Note that the word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example, group theory. There are also "theorems" in science, particularly physics, and in engineering, but they often have statements and proofs in which physical assumptions and intuition play an important role; the physical axioms on which such "theorems" are based are themselves falsifiable.

Theorems in logic

Logic, especially in the field of proof theory, considers theorems as statements (called formulas or well formed formulas) of a formal language. A set of deduction rules, also called transformation rules or a formal grammar, must be provided. These deduction rules tell exactly when a formula can be derived from a set of premises.

Different sets of derivation rules give rise to different interpretations of what it means for an expression to be a theorem. Some derivation rules and formal languages are intended to capture mathematical reasoning; the most common examples use first-order logic. Other deductive systems describe term rewriting, such as the reduction rules for λ calculus.

The definition of theorems as elements of a formal language allows for results in proof theory that study the structure of formal proofs and the structure of provable formulas. The most famous result is Gödel's incompleteness theorem; by representing theorems about basic number theory as expressions in a formal language, and then representing this language within number theory itself, Gödel constructed examples of statements that are neither provable nor disprovable from axiomatizations of number theory.
Terminology

Theorems are often indicated by several other terms: the actual label "theorem" is reserved for the most important results, whereas results which are less important, or distinguished in other ways, are named by different terminology.

1) A proposition is a statement not associated with any particular theorem. This term sometimes connotes a statement with a simple proof.

2) A lemma is a "pre-theorem", a statement that forms part of the proof of a larger theorem. The distinction between theorems and lemmas is rather arbitrary, since one mathematician's major result is another's minor claim. Gauss's lemma and Zorn's lemma, for example, are interesting enough that some authors present the nominal lemma without going on to use it in the proof of a theorem.

3) A corollary is a proposition that follows with little or no proof from one other theorem or definition. That is, proposition B is a corollary of a proposition A if B can readily be deduced from A.

4) A claim is a necessary or independently interesting result that may be part of the proof of another statement. Despite the name, claims must be proved.