What are advantages of the axiomatic method?

Question

In Enderton's Elements of Set theory, the author writes:

The axiomatic method has been useful in other subjects as well as in set theory. Consider plane geometry, for example. It is quite possible to talk about lines and triangles without using axioms. But the advantages of axiomatizing geometry were seen very early in the history of the subject.

What are advantages of the axiomatic method anyway?

Answer 1

Trying to summarize 2 and half millenia of mathematics and logic ...

• John Corcoran, Aristotle's Demonstrative Logic (2009)

• David Hilbert, "Axiomatisches Denken", Mathematische Annalen, 1918

• Hilbert's Program

• Leo Corry, THE ORIGIN OF HILBERT’S AXIOMATIC METHOD

• Hermann Weyl, Philosophy of mathematics and natural science (1st ed., 1949)

• Hermann Weyl, Levels of Infinity : Selected Writings on Mathematics and Philosophy (2013).

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For a "gentle introduction" to axiomatic method, see :

• Alfred Tarski, Introduction to Logic and to the Methodology of the Deductive Sciences (1st English ed., 1941), Ch.VI On the Deductive Method.