Theorem

In mathematics and formal logic, a theorem is a statement that has been proven. The proof of a theorem is a step-by-step explanation that shows the statement is true. It uses special rules to build on facts we already know.

Theorems help us learn important truths in math. They are not just guesses—they are ideas we can be sure about because they have been tested and shown to be true.

In many parts of math, the rules for proving theorems come from systems like Zermelo–Fraenkel set theory or Peano arithmetic. Some big results are called theorems, while smaller ones might be called lemmas, propositions, or corollaries.

The study of theorems and proofs is its own subject, called proof theory. This helps us understand what we can and cannot prove. Important results like Gödel's incompleteness theorems show that some true statements can never be proven in certain systems.

Theoremhood and truth

Before the late 1800s, mathematicians thought theorems were ultimate truths based on simple, clear ideas, like the fact that every natural number has a next number. For example, Euclid's postulates led to proofs such as the idea that the interior angles of a triangle add up to 180°.

But new types of geometries showed that this angle rule only works if certain basic ideas are used. This showed that what seems obviously true can change depending on the starting points chosen. Today, a theorem is seen as something that can be shown to follow from agreed-upon starting points and rules, no matter what those starting points mean in the real world. This helps mathematicians use ideas from one area in totally different areas.

Epistemological considerations

Many math theorems are like "if-then" statements. Their proofs show that if certain conditions are true, then a special result must also be true. This helps us know what we can be sure about in math.

Theorems can be written with symbols, but they are often explained using everyday words to make them easier to read and understand. This way, mathematicians can share their ideas clearly. Theorems are very important in math, and people might call them easy, hard, deep, or even beautiful.

Informal account of theorems

Theorems are important ideas in mathematics that we know are true. They often look like "If A happens, then B will happen." In this case, A is called the hypothesis, and B is the conclusion. For example, a theorem might say, "If a number is even, then half of that number is also a whole number."

Mathematicians usually start with basic ideas called axioms. They use these axioms to prove new theorems. Some theorems are simple and easy to understand, while others are very hard to prove.

Relation with scientific theories

Theorems in mathematics and theories in science are different. Scientific theories can be tested with experiments. They might be wrong if the results don’t match what we expect.

Mathematical theorems are abstract ideas. They are proven using logic and do not need experiments.

Mathematicians sometimes use computers to find patterns and test ideas before proving a theorem. For example, the Collatz conjecture and the Riemann hypothesis are famous problems. These ideas have been checked with many numbers but are still not proven. Even though tests support these ideas, they stay unproven until a logical proof is found.

Terminology

In math, different words describe different kinds of statements. An axiom or postulate is something we start with, accepting it as true without proof. A conjecture is an idea people think might be true but hasn’t been proven yet, like Goldbach's conjecture.

A theorem is a statement that has been proven true using axioms and other theorems. Smaller or simpler truths are called propositions, while important steps in proofs are known as lemmas. Sometimes, a quick result from a theorem is called a corollary. There are also special names for certain types of theorems, like identities, rules, and laws.

Layout

When mathematicians talk about a theorem, they follow a clear way of sharing it. First, they tell what the theorem is and who proved it and when. Then they give the proof, which shows why the theorem is true. The end of the proof is marked with special symbols like "□" or the letters Q.E.D..

Sometimes, smaller ideas called lemmas are shared before the proof to help explain things. After the proof, there might be corollaries. These are results that come right from the theorem. The way these are written can change depending on who is writing or where it is published.

Lore

Many new theorems are proven each year—over a quarter of a million!

There is a famous saying: "A mathematician is a device for turning coffee into theorems." This idea comes from the work of Alfréd Rényi, and is often linked to his friend Paul Erdős, who created many theorems and worked with many other mathematicians.

One very big theorem is the classification of finite simple groups. Its proof is huge—it fills tens of thousands of pages across many articles. People are still trying to make this proof shorter and easier to understand. Another interesting theorem is the four color theorem, which has a computer proof that is very long.

Theorems in logic

In mathematical logic, a formal theory is a group of sentences in a formal language. These sentences have no free variables and are called theorems of the theory. Usually, a theory includes all sentences that follow logically from its starting points.

Theorems in logic can be very important. Some well-known ones include the Compactness of first-order logic, Completeness of first-order logic, and Gödel's incompleteness theorems of first-order arithmetic. These theorems help us understand the limits and properties of logical systems.