What is the axiomatic method?
In maths we cannot prove everything from nothing. So we choose a few statements that look obviously true or that define our subject. These are axioms (or postulates). Then we use logic to prove other statements, called theorems. A proof is a chain: each step follows from axioms or from theorems already proved. This way of building a subject is called the axiomatic method. See Logic and proof for the kinds of proof.
Example 1: Geometry
The oldest example is Euclid's Elements (about 300 BCE), which starts from definitions, common notions and five postulates, such as "a straight line can be drawn between any two points" (see Euclid's geometry). From them he proves theorems, such as the angle sum of a triangle being 180 degrees. In 1899 David Hilbert wrote a stricter list of axioms for geometry. If we change the parallel postulate, we get new, non-Euclidean geometries, such as geometry on a sphere where lines are great circles and the angle sum is more than 180 degrees.
Example 2: Arithmetic
Giuseppe Peano (1889) gave simple axioms for the natural numbers: 0 is a number; every number n has a next number S(n); no number has 0 as its next; different numbers have different next numbers; and induction (if 0 has a property and every n that has it passes it to S(n), all numbers have it). With the rules a + 0 = a and a + S(b) = S(a + b), we can prove 2 + 2 = 4. Write 2 = S(S(0)). Then 2 + 2 = 2 + S(S(0)) = S(2 + S(0)) = S(S(2 + 0)) = S(S(2)) = 4.
Example 3: Probability
In 1933 Andrey Kolmogorov gave three axioms for probability P on a sample space S: (1) P(A) is at least 0; (2) P(S) = 1; (3) if A and B cannot happen together, then P(A or B) = P(A) + P(B). From these we prove, for example, that P(not A) = 1 - P(A). Proof: A and "not A" cannot happen together and between them cover S, so P(A) + P(not A) = P(S) = 1. A further result is P(A) is at most 1, because P(not A) is at least 0. See Events and axioms of probability.
Good axiom systems and models
We want axioms that are consistent (never prove a statement and its opposite), independent (none of them can be proved from the others) and ideally complete (every statement can be settled). In 1931 Kurt Godel showed that for any consistent system strong enough for arithmetic, some true statements cannot be proved from it: this is a limit of the method, not a failure. A model is a thing where all axioms are true: a fair coin is a model of probability, and the surface of a sphere is a model of a non-Euclidean geometry. Using the axioms to describe the real world is modelling (see Mathematical modelling).
Try it
Invent a tiny axiom game: "A friend of a friend is a friend" and "everyone has at least one friend". Prove one small result using only these rules. Then change one rule and see what changes.
Key formulas and definitions
- Axiom: accepted without proof
- Theorem: proved from axioms and earlier theorems
- Kolmogorov: P(A) ≥ 0; P(S) = 1; P(A or B) = P(A) + P(B) if A, B disjoint
- P(not A) = 1 − P(A)
- Peano: a + 0 = a; a + S(b) = S(a + b)
Worked examples
1. Use the probability axioms to find P(not A) if P(A) = 0.3.
P(A) + P(not A) = P(S) = 1. So P(not A) = 1 - 0.3 = 0.7.
2. A and B cannot happen together. P(A) = 0.2 and P(B) = 0.5. Find P(A or B).
By axiom 3, P(A or B) = 0.2 + 0.5 = 0.7.
3. Prove 1 + 1 = 2 from Peano rules, where 1 = S(0) and 2 = S(1).
1 + 1 = 1 + S(0) = S(1 + 0) = S(1) = 2.
Common mistakes
- Thinking axioms are proved. They are chosen and accepted.
- Believing axioms must be "obviously true" in the real world. They only need to be consistent rules.
- Saying a theorem is a guess. A theorem is proved from the axioms.
- Forgetting that the parallel postulate can be changed to give another geometry.