The Axiomatic Method

• Unlike the other kinds of science, math uses a different methodology for establishing mathematical truths, namely, the axiomatic method.

• Axiomatic method was invented by the ancient Greeks and promoted by Euclid, a mathematician working in Alexandria, Egypt around 300 BCE and the author of the second most printed book in the history.

• In "Elements", Euclid showcases the axiomatization of geometry in a rigourous way: he first establishes 5 essential axioms and then proceedes to derive mathematical truths based on them.

• An axiom is the basic assumption considered to be true. However, this description seems loose and downright anti-rigorous, causing much contraversy.

The Axiomatic Contraversy

Every theory must start somewhere; typically, every theory has some object or concept at its core, and it begins by trying to defining its core notion. But the notion excapes definition (e.g. set, number, point, space), so the author is left to informal devices. "Informal definition" is an oxymoron, as is the phrase "formal desciption", especially in mathematical and geographical areas subscribing to the view that the term "description" does not instill the appropriate amount of rigor. Therefore, this stage of theoretical production is a free-flow of embaressing effort of the author to justify concept. Appeals to a reader's intuition are frequently employed. This state of affairs is completely unnecessary. The central notion may be given as a primitive (asmost thing are) or it can beintroduced by oneof the axioms. It has been said that axioms should be self-evident truths, and this stance has also caused a lot of debate. There's no need for that. The axioms should be stated as they are, no justification necessary as long as everyone is familiar with the implication: if the axioms are true, then the derived theorems are necessarily true.