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Using Tarski's axioms or Hilbert's axioms, Euclidean geometry can be described synthetically (in a way that can even be formalized in Coq), i.e. within a theory of first-order logic, or second-order logic if we take an axiom of continuity. I was wondering whether the geometry of conic sections, as studied by Appolonius of Perga for example, can be formulated axiomatically.

Are Tarki's axioms/Hilbert's axioms sufficient to deal with conics/quadrics? If not, what needs to be added? If anyone has any useful reference in mind about the axiomatisation of conic sections, I would also be grateful.

This question is certainly linked to this one, but I have not been able to find a list of axioms that could encompass the geometry of conics systematically (or a reason explaining why the current axiomatisations of Euclidean geometry are enough).

maxbo
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  • "Hilbert's axioms" usually refers to something different from what is formalized in Coq in your link: they refer to a second-order axiomatization that includes a version of the completeness axiom for real numbers (and indeed, Hilbert's axioms in this sense are closely analogous to the axiomatization of the real numbers as a complete ordered field). This axiomatization can only be used inside an ambient set theory, and so it doesn't really make sense to talk about what it can and can't do--it can do everything that the ambient set theory in which you use it can do! – Eric Wofsey Jul 03 '21 at 20:27
  • I think you are right! Once your are within an ambient set theory, then geometry can be done analytically. I should maybe have emphasised on a first-order theory. – maxbo Jul 03 '21 at 21:23

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Using coordinates, pretty much everything in the classical theory of conics can be expressed easily in the first-order theory of the ordered field $\mathbb{R}$: a conic is just the set of ordered pairs $(x,y)\in\mathbb{R}^2$ that satisfy some degree $2$ polynomial equation (and such degree $2$ polynomial equations can themselves be described using finite tuples of real numbers, namely their coefficients). If you fix a pair of oriented perpendicular lines to serve as axes and a unit distance, then Tarski's axioms become bi-interpretable with the complete first-order theory of $\mathbb{R}$ as an ordered field. So, everything about conics can be done using Tarski's axioms.

(OK, not literally everything about conics; there are some non-first order statements about conics. For instance, Tarski's axioms can't talk about the circumference or area of an ellipse since those can only be defined by some limiting process. So, Tarski's axioms can probably do everything that Appolonius did with conics, but not everything that Archimedes did. But there are more basic statements in ordinary Euclidean geometry using only lines that are not first order in a similar way. For instance, Tarski's axioms can't state any general theorems about polygons with arbitrary numbers of sides; they can only state theorems about $n$-gons for fixed values of $n$.)

Eric Wofsey
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  • I see what you mean! The reason I mentioned Tarski's axioms is exactly because they avoid using coordinates and is able to treat geometry (rather a "subset" of Euclidean geometry as you pointed out with the n-gons) in a synthetic/coordinate-free way. Do you know anything about such a program for conics ? – maxbo Jul 03 '21 at 21:27
  • Well, I think one of the big lessons of the modern work on axiomatic geometry is that an analytic/synthetic distinction doesn't hold up--working "synthetically", you can define the field operations and set up all of analytic geometry, and vice versa. Indeed, this is reflected in how Tarski's synthetic axioms are in a rather strong sense equivalent to the first-order theory of the real numbers. – Eric Wofsey Jul 03 '21 at 21:42
  • In any case, though, I wouldn't expect there to be any significant difficulty translating all the arguments that ancient Greeks made about conics into Tarski's axioms in a "natural", synthetic-looking way. I don't see anything about it that would be more difficult than doing the same for Euclid's Elements. – Eric Wofsey Jul 03 '21 at 21:44
  • As far as I know, there's nothing in Apollonius's work that ancient Greeks would have considered beyond the scope of the axiomatic system set up by Euclid. It wasn't some new theory that required new axioms as a foundation; it was just further work in Euclidean geometry. – Eric Wofsey Jul 03 '21 at 21:48
  • Where did you learn the "lessons of the modern work on axiomatic geometry"? I agree that if we need to axiomatise the addition of lengths and area for example, then you can define the field operations and then it's equivalent to doing analytic geometry. However, Tarski's axioms only define congruence and betweeness. How are these enough to define the field operations ? – maxbo Jul 04 '21 at 16:34
  • You can define addition as an operation on congruence classes of line segments (juxtapose two segments to make a longer one). You can define multiplication using sides of similar triangles. I haven't seen the details of how this is done using Tarski's axioms but using Hilbert's axioms it can be found in Chapter 4 of Hartshorne's Geometry: Euclid and Beyond. – Eric Wofsey Jul 04 '21 at 17:03