## Theorem

# Theorem

A theorem (the term is derived from the Greek *theoreo,* which means *I look at* ), denotes either a proposition yet to be proven or a proposition proven correct on the basis of accepted results from some area of mathematics. Since the time of the ancient Greeks, proven theorems have represented the foundation of mathematics. Perhaps the most famous of all theorems is the Pythagorean theorem, which was created by Greek mathematician and philosopher Pythagoras of Samos (c. 582–c. 507 BC).

Mathematicians develop new theorems by suggesting a proposition based on experience and observation that seems to be true. These original statements are only given the status of a theorem when they are proven correct by logical deduction. Consequently, many propositions exist that are believed to be correct but are not theorems because they cannot be proven using deductive reasoning alone.

## Historical background

The concept of a theorem was first used by the ancient Greeks. To derive new theorems, Greek mathematicians used logical deduction from premises they believed to be self-evident truths. Since theorems were a direct result of deductive reasoning, which yields unquestionably true conclusions, they believed their theorems were undoubtedly true. Early mathematician and philosopher Thales (640–546 BC) suggested many early theorems and is typically credited with beginning the tradition of a rigorous, logical proof before the general acceptance of a theorem. The first major collection of mathematical theorems was developed by Greek mathematician Euclid of Alexandra (c. 325–c. 265 BC) around 300 BC, in a book called *The Elements*.

The absolute truth of theorems was readily accepted up until the eighteenth century. At this time mathematicians, such as Karl Friedrich Gauss (1777–1855), began to realize that all of the theorems suggested by Euclid could be derived by using a set of different premises, and that a consistent non-Euclidean structure of theorems could be derived from Euclidean premises. It then became obvious that the starting premises used to develop theorems were not self-evident truths. They were, in fact, conclusions based on experience and observation—and not necessarily true. In light of this evidence, theorems are no longer thought of as absolutely true. They are only described as correct or incorrect based on the initial assumptions.

## Characteristics of a theorem

The initial premises on which all theorems are based are called axioms. An axiom, or postulate, is a basic fact that is not subject to formal proof. For example, the statement that there is an infinite number of even integers is a simple axiom. Another is that two points can be joined to form a line. When developing a theorem, mathematicians choose axioms, which seem most reliable based on their experience. In this way, they can be certain that the theorems are proved as near to the truth as possible. However, absolute truth is not possible because axioms are not absolutely true.

To develop theorems, mathematicians also use definitions. Definitions state the meaning of lengthy concepts in a single word or phrase. In this way, when people talk about a figure made by the set of all points which are a certain distance from a central point, one can just use the word *circle*.

*See also* Symbolic logic.

### KEY TERMS

**Axiom** —A basic statement of fact that is stipulated as true without being subject to proof.

**Deductive reasoning** —A type of logical reasoning that leads to conclusions which are undeniably true if the beginning assumptions are true.

**Definition** —A single word or phrase that states a lengthy concept.

**Pythagorean theorem** —An idea suggesting that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. It is used to find the distance between two points.

## Resources

### BOOKS

Burton, David M. *The History of Mathematics: An Introduction*. New York: McGraw-Hill, 2007.

Dunham, William. *Journey Through Genius: the Great Theorems of Mathematics*. New York: Wiley, 1990.

Kline, Morris. *Mathematics for the Nonmathematician*. New York: Dover, 1967.

Lloyd, G. E. R. *Early Greek Science: Thales to Aristotle*. New York: W. W. Norton, 1970.

Newman, James R., ed. *The World of Mathematics*. New York: Simon and Schuster, 1956.

Paulos, John Allen. *Beyond Numeracy*. New York: Knopf, 1991.

Setek, William M. *Fundamentals of Mathematics*. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.

Solow, Daniel. *How to Read and Do Proofs: An Introduction to Mathematical Thought Processes*. New York: Wiley, 2002.

Sundstrom, Theodore A. *Mathematical Reasoning: Writing and Proof*. Upper Saddle River, NJ: Prentice Hall, 2003.

Perry Romanowski

## Theorem

# Theorem

A theorem (the term is derived from the Greek *theoreo*, which means *I look at*) denotes either a proposition yet to be proven, or a proposition proven correct on the basis of accepted results from some area of **mathematics** . Since the time of the ancient Greeks, proven theorems have represented the foundation of mathematics. Perhaps the most famous of all theorems is the **Pythagorean theorem** .

Mathematicians develop new theorems by suggesting a proposition based on experience and observation which seems to be true. These original statements are only given the status of a theorem when they are proven correct by logical deduction. Consequently, many propositions exist which are believed to be correct, but are not theorems because they can not be proven using deductive reasoning alone.

## Historical background

The concept of a theorem was first used by the ancient Greeks. To derive new theorems, Greek mathematicians used logical deduction from premises they believed to be self-evident truths. Since theorems were a direct result of deductive reasoning, which yields unquestionably true conclusions, they believed their theorems were undoubtedly true. The early mathematician and philosopher Thales (640-546 b.c.) suggested many early theorems, and is typically credited with beginning the tradition of a rigorous, logical **proof** before the general acceptance of a theorem. The first major collection of mathematical theorems was developed by Euclid around 300 b.c. in a book called *The Elements*.

The absolute truth of theorems was readily accepted up until the eighteenth century. At this time mathematicians, such as Karl Friedrich Gauss (1777-1855), began to realize that all of the theorems suggested by Euclid could be derived by using a set of different premises, and that a consistent non-Euclidean structure of theorems could be derived from Euclidean premises. It then became obvious that the starting premises used to develop theorems were not self-evident truths. They were in fact, conclusions based on experience and observation, and not necessarily true. In **light** of this evidence, theorems are no longer thought of as absolutely true. They are only described as correct or incorrect based on the initial assumptions.

## Characteristics of a theorem

The initial premises on which all theorems are based are called axioms. An axiom, or **postulate** , is a basic fact which is not subject to formal proof. For example, the statement that there is an infinite number of even **integers** is a simple axiom. Another is that two points can be joined to form a line. When developing a theorem, mathematicians choose axioms, which seem most reliable based on their experience. In this way, they can be certain that the theorems are proved as near to the truth as possible. However, absolute truth is not possible because axioms are not absolutely true.

To develop theorems, mathematicians also use definitions. Definitions state the meaning of lengthy concepts in a single word or phrase. In this way, when we talk about a figure made by the set of all points which are a certain **distance** from a central point, we can just use the word *circle*.

See also Symbolic logic.

## Resources

### books

Dunham, William. *Journey Through Genius.* New York: Wiley, 1990.

Kline, Morris. *Mathematics for the Nonmathematician.* New York: Dover, 1967.

Lloyd, G.E R. *Early Greek Science: Thales to Aristotle.* New York: W. W. Norton, 1970.

Newman, James R., ed. *The World of Mathematics.* New York: Simon and Schuster, 1956.

Paulos, John Allen. *Beyond Numeracy.* New York: Knopf, 1991.

Perry Romanowski

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Axiom**—A basic statement of fact that is stipulated as true without being subject to proof.

**Deductive reasoning**—A type of logical reasoning that leads to conclusions which are undeniably true if the beginning assumptions are true.

**Definition**—A single word or phrase that states a lengthy concept.

**Pythagorean theorem**—An idea suggesting that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. It is used to find the distance between two points.

## theorem

**theorem**
•minimum • maximum • optimum
•**chrysanthemum**, helianthemum
•cardamom • Pergamum • sesamum
•per annum • magnum • damnum
•**Arnhem**, Barnum
•**envenom**, venom
•interregnum • Cheltenham • arcanum
•**duodenum**, plenum
•platinum • antirrhinum • Bonham
•summum bonum • Puttnam
•ladanum • molybdenum • laudanum
•**origanum**, polygonum
•organum • tympanum
•**laburnum**, sternum
•gingham • Gillingham • Birmingham
•Cunningham • Walsingham
•Nottingham • wampum • carom
•Abram • panjandrum • tantrum
•angstrom • alarum • candelabrum
•**plectrum**, spectrum
•**arum**, harem, harum-scarum, Sarum
•**sacrum**, simulacrum
•maelstrom • cerebrum • pyrethrum
•Ingram
•**sistrum**, Tristram
•Hiram
•**grogram**, pogrom
•**nostrum**, rostrum
•**cockalorum**, decorum, forum, jorum, Karakoram, Karakorum, Mizoram, pons asinorum, quorum
•wolfram • fulcrum • Durham
•conundrum • buckram • lustrum
•**serum**, theorem
•labarum • marjoram • pittosporum
•Rotherham • Bertram

## theorem

the·o·rem / ˈ[unvoicedth]ēərəm; ˈ[unvoicedth]i(ə)r-/ • n. Physics & Math. a general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths. ∎ a rule in algebra or other branches of mathematics expressed by symbols or formulae. DERIVATIVES: the·o·re·mat·ic / ˌ[unvoicedth]ēərəˈmatik; ˌ[unvoicedth]i(ə)rə-/ adj.

## theorem

**theorem** general proposition demonstrable by argument. XVI. — F. *théorème* or late L. *theōrēma* — Gr. *theṓrēma* speculation, theory, proposition to be proved, f. *theōreîn* look at, f. *theōrós* spectator (see THEORY).

So **theoretic(al)** †speculative, contemplative; pert to theory. XVII. — late L. *theōrēticus* — Gr. *theōrētikós*. Hence **theoretician** XIX.

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