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I have some course in mathematics. Group theory, Ring theory, topology and etc. All of this theory begin with axioms. Whether every theory in mathematics should getting started with axioms? Is axiomatic method the only way to introduce a mathematical theory? If not, is there an example? thanks.

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    I don't like the terminology 'axioms of a group'. You can give a definition of a group and then you can explicitly verify that $\mathbb{R},+$ satisfies this definition. However, you can also start asking what $\mathbb{R}$ is, or even what a set is. At some point you will have to stop asking what something is and take that thing as a given. You could axiomatize the thing you need and work from there. But this is completely different from the axioms of a group, which is just a definition. – 2017-01-20
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    what about Euclidean geometry? @Mathematician42 – 2017-01-20
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    Depends on how you want to do it. You can work entirely with coordinates and simply use algebra to solve problems. In this case you'll need the same 'axioms' as for usual set theory. If you want to give synthetic proves (often on pictures) you need specify certain rules. For example, two lines (whatever that might be ) either coincide, intersect in one point or have no points in common. Obviously, you'll want that these axioms coincide with the analogous statements in the more algebraic setting. Look up the axioms of Euclidean geomtery. These are the simple rules you need. – 2017-01-20
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    A very similar question http://math.stackexchange.com/q/1865687/269624 – 2017-01-20
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    No; it is not. Arithmetic is here "from the beginning" of mathematics but was axiomatized only towards the end of 19th Century. – 2017-01-20
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    Cantor's set theory has been axiomatized only **after** the discobery of the paradoxes. – 2017-01-20
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    Calculus was developed duirng 17th Century in a non-axiomatized way. – 2017-01-20
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    WikiMapia says : "Combinatorics is an example of a field of mathematics which does not, in general, follow the axiomatic method." Learn more here https://groups.google.com/d/msg/sci.math/lT3KmcLURBs/0WwsSliPAgAJ – 2017-12-21
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    @MauroALLEGRANZA I think the OP is concerned with theory introduction not development – 2017-12-21

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