Welcome to BookofProofs

1. Assert the truth of one or more statements (and call them axioms or postulates). Add the postulates to a list (and call that list theory).
2. Given all statements in your theory, logically derive new statements which are true (and call them propositions or theorems).
3. Add the newly derived theorems to your theory.
4. Continue with step 2.

The method is very powerful. It works like a snowball and allows constructing complex theories from easy to understand basic axioms.

A Simple Example ("Little Bird Theory"):

1. Axiom 1: "All ravens are black". Axiom 2: "Every bird is a raven". In its first iteration, our "Little Bird Theory" consists of the axioms 1 and 2.
2. Statement 1: All birds are black. Proof: Let a thing be a bird. According to Axiom 2, the thing is a raven. According to Axiom 1, the thing must be black.
3. After this proof, our "Little Bird Theory" became bigger. Now, it consists of the axioms 1 and 2, including the statement 1.
4. We could now continue with step 2:
   • Statement 2: White things are not birds. Proof: Let a thing be white. According to Statement 1, the thing cannot be a bird.
   • After this proof, our "Little Bird Theory" became even bigger. Now, it consists of the axioms 1 and 2, including the statements 1 and 2.

The "Little Bird Theory" is nonsense, since it disagrees with our daily experience. But apart from some technical details, which are unimportant here (eg. we haven't defined how exactly we derive new theorems using logical steps), there is nothing to complain about this theory.

The success of the mathematics, physics and computer sciences based on the axiomatic method lies in the clever choice of axioms at the very beginning of a new theory. The better the choice of axioms, the more likely the axiomatic method will produce a theory which has better applications in the real world or makes better predictions about the real world. For instance, the axiom "The speed of light is constant whether the ray be emitted by a stationary or by a moving body" allowed Albert Einstein at the beginning of the 20th century to develop his Theory of Relativity. Now, we harvest this theory by applications like the GPS system, which wouldn't work, if the axiom was wrong.