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We can think of Euclidean Geometry and Cartesian (Coordinate) Geometry as equivalent, in the sense that some proposition is true in Euclidean Geometry iff it's true in Coordinate Geometry. It makes sense to me to think of Coordinate Geometry as Euclidean Geometry "arithmetized", or even as a model of Euclidean Geometry. I'm comfortable with it because we study synthetic (axiomatic) Euclidean Geometry then we study Cartesian Geometry.

But the situation may change when it comes to Projective Geometry, I think most people get to know Coordinate Projective Geometry (e.g. projective spaces) before Synthetic Projective Geometry (I even think that a lot of people don't know that such thing as Synthetic Projective Geometry exists).

Now my question:

Does it help to know the "synthetic" version of a geometry when we use a Coordinate Version (Model) of it in more advanced math? (Not only projective, but affine, hyperbolic...)

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The synthetic viewpoint is more general!

This is a benefit since stating things abstractly in terms of primitive notions is probably easier to grasp rather than delving immediately into coordinates. Compare:

  1. Two lines intersect in at most one point

  2. The system $\{ax+by+c=0;dx+ey+f=0\}$ has at most one solution.

The first comes with less baggage than the second :) You can imagine similar things like the difference between the abstract definition of a linear transformation and their representation in terms of matrices.

For another thing, generality just captures more things in the same net.

Coordinatization is a great example. A necessary and sufficient condition for an affine plane to be coordinatizable by a division ring (this means "exhibited as $D^2$ for a division ring $D$ in "coordinate geometry") is that the plane satisfy Desargues' Theorem. The same is true for projective planes with the projective version of Desargues' Theorem. Naturally, this means that there are projective and affine planes that cannot be exhibited as vector spaces over a division ring.

Is it still possible to coordinatize projective planes with something more general than a division ring? Yes, sometimes! This is what planar ternary rings accomplish for projective planes satisfying certain properties.

So you can see, while modeling geometries in $F^n$ for a field $F$ is concrete and useful, there are still lots of other synthetic planes that you can prove many of the same theorems in, and yet they may not have such a representation like $F^n$.

It's important to remember that when Euclid did geometry, he did not have the real numbers, and he certainly wasn't thinking about it in terms of $\Bbb R^n$. Euclid's approach was all based on congruence, but not measurment with real numbers. This transition to "real metric geometry" is a relatively new method of teaching Euclidean geometry.

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  • $\begingroup$ What do you mean by coordinatization ? I think what you understand by coordinatization is something broader than I think. $\endgroup$
    – Mandero
    Jul 23, 2014 at 15:37
  • $\begingroup$ @Mandero It may be used in different ways elsewhere, but here I mean exactly what I described: assigning points to elements of $D^n$ in such a way that the lines of the geometry are lines in $D^n$. It's just "doing Cartesian analytic geometry in $D^n$." $\endgroup$
    – rschwieb
    Jul 23, 2014 at 16:20

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