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Do we have any axioms that allow us represent Cartesian coordinates on a graph in euclidean space or is it purely intuitive? It's easy to intuitively justify where $(0,1)$ and $(1,1)$ would lie in space in relation to the origin and to each other. Is the Cartesian Coordinate system just a tool to intuitively represent real or complex, ect. functions? Or can we build it from axioms?

Is the calculus of $\mathbb R$ a model for Cartesian/Euclidean space? Can we fit real analysis into the axioms of Euclid or say Hilbert's (foundational) geometry? And if so how do we jump from Euclidean geometry where points and lines, etc. don't have actually "positions" in space to Cartesian coordinates where they do?

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  • $\begingroup$ We have axioms for plane geometry and we have axioms of the real number field. Then we verify that $\mathbb R^2$ is a model of plane geometry. See D.Hilbert, Foundations of Geometry, page 17-18. $\endgroup$ – Mauro ALLEGRANZA Oct 4 at 7:32
  • $\begingroup$ See very similar post. $\endgroup$ – Mauro ALLEGRANZA Oct 4 at 7:51
  • $\begingroup$ At the very bottom of things, of course, the axioms themselves are supported by intuition. $\endgroup$ – David K Oct 4 at 13:20

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