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What distinguishes maths? Mathematical method 1. Mathematics is a FORMAL system of knowledge. 2. The foundation of maths is AXIOMS. 3. If you apply RULES OF INFERENCE to the axioms, you create mathematical knowledge, called THEOREMS. 2. The foundation of maths is AXIOMS. Axioms are fundamental laws. Axioms are obtained by reflecting rationally on what we know. They are examples of a priori knowledge, accepted by all mathematicians. Mathematics is based on axioms Many mathematicians accept them as given, they know they are the foundation of maths but would find them hard to define. Examples of axioms Euclid's elements Non-euclidean geometry Seven axioms of set theory (ZermeloFrankel) Peano's Postulates Peano's Postulates Define the fundamental laws of the number system These are the fundamental rules of arithmetic Peano says there are such things as numbers and they can be defined by 5 axioms 1. 0 is a number 2. Every number has at least one and at most one successor which is a number 3. 0 is not the successor of any number 4. No two numbers have the same successor 5. Whatever is true of 0, and is also true of the successor of any number when it is true of that number, is true of all numbers If you apply RULES OF INFERENCE to the axioms, you create mathematical knowledge, called THEOREMS. Inference is the forming of conclusions from the information available Rules of inference are those rules which mathematicians apply deductively to the mathematical information available to them, the axioms If....then.... A well-known rule of inference An arithmetical example: if 1+6=7 and 5+4=9 then (1+6) + (5+4) = 7 + 9 An algebraic example If x=y and p=q then x+p = y+q Theorems A statement created by deductively applying the rules of inference to axioms A statement of mathematical knowledge Example: Pythagoras Theorem The square on the hypotenuse is equal to the sum of the squares on the other 2 sides This theorem is based on the 10 axioms of Euclid's Elements (but mathematicians take these as given) Are we happy? Does it all seem a bit vague? Are we relying on “givens”? Where have the axioms come from? Is this why we have proof? The mathematical method a deductive process Define axioms Then apply rules of inference Then create theorems Then apply rules of inference Then create more theorems Then apply rules of inference Then create even more theorems and so on ad infinitum....