Hilbert's Foundations of Geometry is obviously part of this "equation" as well. --- As I may have written before, I am more and more becoming intrigued and thrilled by the thought what foundations (both for mathematics and for computing) today would look like, had category theory been created, and would have been established even in rudimentary forms, already during the times Frege and Peano debated, i.e., categorical language would also have influenced Principia. By the time we come to the 1920's, efforts to adopt the Liar and similar paradoxes in mathematical "proof" would probably have been rejected, i.e., Gödel's "Theorems" would probably have been viewed as paradoxes (which, in my view, they essentially are), and those paradoxes would probably have been solved by now. A "categorization" of Hilbert's Foundations of Geometry might not provide all that much new insight to foundations, but a categorization of Hilbert's and Bernays' Grundlagen der Mathematik I and II probably will, obviously including a critical and categorical examination of Gödel's "technique". Doing so, I feel that the Foundations of Mathematics would be brought up to an entire new level of investigation, and indeed with the support of category theory. --- Best, Patrik On 2026-06-27 16:00, John Baez via Categories wrote:
Hi -
Geometry in the style of Euclid is called "synthetic" in distinction to using coordinates à la Dedekind, but I haven't heard coordinate geometry being called "analytic" and I'm not sure what Euclid was "putting together".
I have always heard "analytic geometry" used to mean coordinate geometry. From the Wikipedia article Analytic geometry [5]:
In mathematics [1], analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry [2] using a coordinate system [3]. This contrasts with synthetic geometry [4].
Best, jb
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Links: ------ [1] https://en.wikipedia.org/wiki/Mathematics [2] https://en.wikipedia.org/wiki/Geometry [3] https://en.wikipedia.org/wiki/Coordinate_system [4] https://en.wikipedia.org/wiki/Synthetic_geometry [5] https://en.wikipedia.org/wiki/Analytic_geometry