What is the importance of axiomatic geometry? I understand that analytic geometry is really helpful for some kinds of geometries. And in most instances it's all we need.

Is the axiomatic approach used to describe "weird" geometries--ones that we cannot describe or embed analytically? What are some examples of such geometries and what are some applications?

  • $\begingroup$ It's the most direct heritage we have from Euclid. So there is the cultural importance. $\endgroup$
    – Arthur
    Oct 19, 2019 at 21:37
  • $\begingroup$ I see. Are there any axiomatic geometries that cannot be expressed analytically? $\endgroup$
    – BENG
    Oct 19, 2019 at 21:52

1 Answer 1


Axiomatic/synthetic geometry is the most intuitive way to introduce the concept of building up complicated ideas from simpler ones without relying on dense algebraic proofs which intimidate people—you just “construct” new objects

It’s historically super important; I think Euclid’s Elements is the most-read book after the Bible due to its role in historical education.

Through Hilbert, it helped found modern mathematics and abstract geometry.

It also lets us work ideas that may be difficult to express analytically in a simple, intuitive form.

I would recommend checking out the game Euclidia for an elegant approach to axiomatic synthetic geometry. It expresses the feeling of playing around with geometric objects to make new ones. This was ultimately the root of our modern proof-based approach to Mathematica.

Finally, most historical Western math up until the 1700s is grounded in axiomatic synthetic geometry. For example, Newton’s mathematic writing all rests on Euclidean geometry.


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