Theorem

In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to show that the theorem follows from axioms and previously proven ideas.

Theorems are important because they help us understand deep truths in mathematics. Unlike guesses or ideas, theorems are statements we can be certain about because they have been carefully checked and shown to be true.

In many areas of math, the rules used to prove theorems are based on well-known systems like Zermelo–Fraenkel set theory or Peano arithmetic. Some big results are called theorems, while smaller helpful results might be called lemmas, propositions, or corollaries.

The study of theorems and proofs has become a subject itself, called proof theory. This area helps us understand the limits of what we can prove, as shown by important results like Gödel's incompleteness theorems. These show that in any system that includes basic number facts, there will always be true statements that cannot be proven within that system.

Theoremhood and truth

Before the late 1800s, mathematicians believed that theorems were ultimate truths based on self-evident 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°.

However, 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 mathematical theorems are like "if-then" statements. Their proofs show that if certain conditions, called hypotheses or premises, are true, then a particular conclusion must also be true. This helps us understand what we can be certain about in mathematics.

Theorems can be written using symbols, but they are often explained in everyday language to make them easier to read and understand. This way, mathematicians can share their ideas clearly and convincingly. Theorems are very important in mathematics, and people may describe them as easy, hard, deep, or even beautiful, depending on how they see them.

Informal account of theorems

Theorems are important ideas in mathematics that have been proven to be 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 and can surprise mathematicians by connecting different areas of math.

Relation with scientific theories

Theorems in mathematics and theories in science are different. Scientific theories can be tested through experiments and might be proven wrong if the results don’t match predictions. In contrast, mathematical theorems are abstract statements that 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 that have been checked for many numbers but are still not proven. Even though tests support these ideas, they remain 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 share a theorem, they usually follow a clear format. First, they state the theorem, often mentioning who proved it and when. Then comes the proof, which explains why the theorem is true. The end of the proof is marked with special symbols like "□" or the letters Q.E.D., short for "quod erat demonstrandum," meaning "which was to be shown."

Sometimes, definitions or smaller ideas called lemmas are shared before the proof to help explain the theorem. After the proof, there might be corollaries, which are results that follow directly from the theorem. The way these are written can vary depending on the author or the publication's style.

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 while enjoying coffee.

One very big theorem is the classification of finite simple groups. Its proof is huge—it fills tens of thousands of pages across 500 articles written by about 100 authors! 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 too long for any person to read all the way through.

Theorems in logic

In mathematical logic, a formal theory is a collection 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.