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Euclid formulated 5 postulates for what we now call Euclidean geometry.

However, modern university level mathematics no longer develop mathematics on the basis of such geometric axioms.

Rather, they start out with basic concepts such as fields, vector spaces, topological spaces, affine spaces, metrics, manifolds, etc, and then(among other things) build geometric systems on the basis of that.

This seems to me to be even very different from 19th century re-axiomatizations of Euclidean geometry.

How is Euclid's axiomatization of geometry connected to the modern system? How do we reformulate Euclid's (or 19th century versions) in terms of modern notions such as affine spaces, metrics, etc?

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    See [Hilbert's axiomatization of geometry](https://en.wikipedia.org/wiki/Hilbert's_axioms). – 2017-02-26
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    See also [Foundations of geometry](https://en.wikipedia.org/wiki/Foundations_of_geometry) with ref to [Birkhoff's axioms](https://en.wikipedia.org/wiki/Foundations_of_geometry#Birkhoff.27s_axioms). – 2017-02-26
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    See also [Tarski's axioms](https://en.wikipedia.org/wiki/Tarski%27s_axioms). – 2017-02-26
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    My question was a bit different from what I think it seemed. I've reformulated it. – 2017-02-27
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    You can see Francis Borceux, [Geometric Trilogy](http://www.springer.com/la/book/9783319018041), 3 vols. – 2017-02-27
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    See also the post [what-is-the-modern-axiomatization-of-euclidean-plane-geometry](http://math.stackexchange.com/questions/80930/what-is-the-modern-axiomatization-of-euclidean-plane-geometry). – 2017-02-27
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    For a modern formalization of Euclid's axiom see https://arxiv.org/abs/0810.4315 and https://arxiv.org/abs/1710.00787 – 2018-05-25

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