Why not axiomize Euclidean Geometry with coordinates? Hilbert attempted to formalize geometry with his axiom system which is somewhat complex, and it isn't obvious which axioms can be thrown away or if any are being implicitly assumed. I present an alternative axiomatic system, I want to know if it has any serious problems.


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*Assume the construction of the real numbers.

*A point is defined to be any element of $\mathbb{R}^n$ .

*A line in $\mathbb{R}^2$ is defined to be $\{(x,y):ax+by+c=0\}$ for real constants $a,b,c$.

*$\mathbb{R}$ is a line.

*A plane in $\mathbb{R}^3$ is defined to be $\{(x,y,z):ax+by+cz+d=0\}$ for real constants $a,b,c,d$.

*$\mathbb{R}^2$ is a plane.

*The distance between two points $x,y\in\mathbb{R}^n$, $x=(x_1,x_2\dotsb x_n)$, $y=(y_1, y_2\dotsb y_n)$ is defined to be $\sqrt{(x_1-y_1)^2+(x_2-y_2)^2\dotsb(x_n-y_n)^2}$

*$|P|$ is the distance between $P$ and $(0,0,\dotsb 0)$

*If $x = (x_1,x_2\dotsb x_n)$ and $y = (y_1, y_2\dotsb y_n)$, their sum is defined as $x+y=(x_1+y_2,x_2+y_2,\dotsb x_n+y_n)$.

*For a point $x$ and real constant $k$, their product is defined as $kx = (kx_1,kx_2,\dotsb kx_n)$

*The cross and dot products are defined as usual with the coordinate definitions.

*The sine and cosine function are defined with the series. Other trigonometric functions are defined as ratios and inverses of those.

*For points $A,B,C$, $A\not = B$, $A\not = C$, the angle $\angle BAC$ is defined to be $\arccos\left(\frac{(B-A)\cdot(C-A)}{|B-A||C-A|}\right)$ (this is just a restatement of the dot product geometric interpretation).


I think from this all of Euclidean geomtry follows.
 A: First, let me point out that this has indeed been done before - there's nothing inherently wrong with the approach.
Now, some comments.
First, note that you're writing these axioms assuming properties of $\mathbb{R}$. That is, you haven't really axiomatized geometry - rather, you've reduced it to the task of axiomatizing $\mathbb{R}$. This in turn suggests the problem of constructing $\mathbb{R}$ (what exactly is a real number in the first place?), which is a surprisingly difficult task.
Meanwhile, another issue is more philosophical in flavor: are we as happy with an axiomatization of geometry that relies on non-geometric ideas? While this is undeniably useful, one might reasonably want to axiomatize geometry on its own terms - that is, without reference to any non-geometric concepts. (Arguably the algebraic structure of $\mathbb{R}$ is geometric - this isn't really a precise distinction - but whatever.) This desire makes a lot of sense given the old view of geometry and algebra as fundamentally different subjects, and also from the modern point of view of mathematical logic that simpler axiomatizations often have nice properties which let us prove things about the objects they axiomatize.
Basically, when thinking about axiomatizing something, there are a couple questions you should keep in mind:


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*What are you taking for granted? E.g. are you using some other structure (like the reals) in your axiomatization?

*What is your goal - philosophically, mathematically, maybe even aesthetically? There is no such thing as a "best" axiomatization: different purposes call for different tools.
