What is the difference between axiom, theory and theorem?

(Juliana Caro/)

There are three things that happen in the science of mathematics. An axiom is a fact that is not questioned. It should not be understood as a fundamental truth in life, but rather as an unbreakable rule when formulating a theorem. For example, Euclidean geometry (which encompasses many formulas and concepts you see in high school) only works if you take five axioms for granted, such as that it is always possible to draw a straight line between two points. Theorem is the result of something that you can prove with logical steps from axioms.

For example, the Pythagorean Theorem says that “the sum of the squares of the legs is equal to the square of the hypotenuse” – but this is only true if you take into account the axioms of Euclidean geometry. Unlike axioms, theorems can be questioned, as they are more subject to interpretation. In science in general, “theory” is the same thing as theorem in mathematics, namely: a logical conclusion drawn from the facts and which can be questioned.

But in mathematics “theory” is more understood as a field of study. For example: Euclidean geometry is a theory assembled from axioms and within which the Pythagorean Theorem can be proved.

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