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I read somewhere that most areas of math don't start out with axioms. They become formulated later on.

Are there any areas of math, in which that area was first developed and expanded from axioms?

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I doubt it, but to prove it, I'd have to know every area of Mathematics, and the history of its development (and I don't). –  Gerry Myerson Mar 11 '13 at 1:39
    
To add on what @Gerry said, this is not how mathematicians work. First you come up with a working notio ln, play with it a bit, and then you distill its formal properties as axioms. I seriously doubt there is any particular field that has such a... backwards history. –  Asaf Karagila Mar 11 '13 at 1:48
    
Well wait what about Category Theory? That seems like it would naturally begin with a set of axioms. –  levitopher Mar 11 '13 at 1:49
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@cduston: category theory began (historically) with the existing examples of many categories: groups, rings, sets, vector spaces, etc. Then the axioms were distilled from the examples. This is the way things usually work in math - examples come first, axioms second. Of course, formally, we prove things by starting with axioms. But the axioms themselves come from considering a class of examples and trying to isolate the relevant properties. –  Carl Mummert Mar 11 '13 at 1:53
    
And to add on @Carl's comment: this is why we have so many "normal" or "regular" objects in mathematics. –  Asaf Karagila Mar 11 '13 at 2:01

2 Answers 2

The way things usually work in math is that examples come first, and axioms second. Of course we prove things formally by starting with axioms. But the axioms themselves are found by considering a class of examples and trying to isolate the relevant properties.

So, although it would be possible in principle to start a new area of mathematics by arbitrarily choosing a set of axioms, that is not the way that mathematical research is actually done.

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Here is what happens when an area is first developed and expanded from axioms.

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