Mathematics

Questions tagged [axiomatic-geometry]

Questions about axiomatic systems for geometry. Use this tag if you're looking for a proof starting directly from some set of axioms (e.g., Hilbert's axioms for Euclidean geometry), or if you have a question about the axioms themselves.

Historically, Euclid's Elements were the first attempt to put geometry (or any other branch of mathematics) on a rigorous footing, proving theorems starting from a fixed group of postulates and common notions.

Since then, many mathematicians have asked questions such as:

  • Are all the axioms used by Euclid really necessary, or can some of them (notably the parallel postulate) be proven from the other axioms?
  • What sort of objects satisfy some or all of the axioms of geometry?
  • What collections of axioms can completely specify Euclidean geometry?
  • What sort of geometries do we obtain, and which theorems can we still prove, by dropping or negating some of these axioms?

Such questions are the domain of axiomatic geometry.

More modern axiomatizations of geometry include Hilbert's axioms (which are possibly the most widely used today) as well as other systems proposed by Birkhoff and Tarski.

In addition to Euclidean geometry, objects studied in axiomatic geometry include hyperbolic geometry, elliptic geometry, and even finite incidence structures.

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Why is "lies between" a primitive notion in Hilbert's Foundations of Geometry?

I read this question: Hilbert's Foundations of Geometry Axiom II, 1 : Why is this relevant? and its answer by Eric Wofsey. In this answer, it is stated that "lies between" a primitive notion in Hilbert's Foundations of Geometry. Why is it a…
wythagoras
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How axioms of inner product ensure that an instantiation/realization capture notion of angle correctly?

Axiomatic definition of inner product can lead to various instantiations like euclidean inner product or complex inner product or weighted inner product etc. Whatever the special case, we can be sure, $\langle x, x \rangle$ will be candidate for…
irman
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Prove: "if three points are on a straight line, at least one point is between the other two."

In Wikipedia, the third order axiom of Hilbert's axioms states that "Of any three points situated on a line, there is no more than one which lies between the other two. Note : The existence part ("there is at least one") is a theorem. So if there…
user656338
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Is it possible to take S.S.S. Congruence criterion as a postulate and prove S.A.S. and A.S.A. through it?

In all of the treatments of elementary Euclidean geometry which I've seen so far, the section about triangle congruences introduces S.A.S. criterion as the basic postulate from which A.S.A. and S.S.S. criteria are deduced. I remember reading…
David06
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Book recommendation for Axiomatic three-dimensional geometry and analytic geometry

I'm reading John Lee's book Axiomatic Geometry and I enjoy it a lot. It includes a detailed treatment of Euclidean plane geometry with rigorous proofs from axioms. I'm looking for books about Euclidean three-dimensional geometry and Cartesian…
yh05
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A confusion about the second connection axiom of Euclidean Geometry

In the book of Foundations of Geometry by Hilbert, at page 2, it is stated that I, 1. Two distinct points A and B always completely determine a straight line a. We write AB = a or BA = a. I, 2. Any two distinct points of a straight line completely…
Our
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Can we really intersect circles?

Euclid's Elements were a great work, but in modern standards it is not totally rigorous. One of its biggest flaws is right on Proposition 1, where he doesn't prove that the two circles intersect, in order to build an equilateral triangle. The…
J. C.
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Question about Pasch's Postulate, line going through all three sides of a triangle

I've been reading the textbook Elementary Geometry from an Advanced Standpoint by Edwin E. Moise (3rd ed.). My problem with his wording of Pasch's Postulate, and then a subsequent problem which aggravates my confusion. To start, this is his exact…
weirdesky
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How to define orientation from the axioms of Euclidean geometry

My question originates from this answer by Kulisty , where it is stated that the notion of orientation can be defined using only the axioms of neutral geometry . I am particularly interested in proving the transitivity of that orientation relation.…
Antonio
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Trouble proving that a plane in (synthetic) projective space containing two points must contain the line between them

I'm working through Hartshorne's Foundations of Projective Geometry , and have hit a sticking point with problem 7(a). For clarity, I'll reproduce the axiom system presented for projective 3-space: Two points determine a unique line. Three…
Fargle
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Using another congruence criterion as axiom in Hilbert's axiomatic geometry

I have a question about Hilbert's axiomatic system. In this system we have the SAS (Side-Angle-Side) congruence criterion as an axiom, and then we prove the SSS (Side-Side-Side) and ASA (Angle-Side-Angle) criteria as theorems. I was wondering then…
IlaRossi
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Model of ordered plane which is neither isomorphic to $\mathbb{R}^2$ nor to Klein model

Let $B_{\mathbb{R}}\subset\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ be standard (strict) betweenness relation on $\mathbb{R}$ i.e. $$B_{\mathbb{R}}(abc):\iff\left(a<b<c \vee c<b<a\right)$$ My definition of ordered plane is as follows: $P$ is a set…
Kulisty
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Playfair's Axiom for parallel planes instead of lines

In three-dimensional space, can Playfair's Axiom: Given a line $a$ and a point $P$ not in $a$, there is at most one line in $P$ parallel to $a$. be “replaced by“ the following axiom? Given a plane $\alpha$ and a point $P$ not in $\alpha$, there is…
Larry
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How does Hilbert's axiomatization relate to set theory?

I'm studying Hilbert's axiomatization of Euclidean geometry, and I'm trying to combine my current understanding into my knowledge on mathematical logic (not very much). At the beginning of this axiomatic system, it states the undefined terms are…
Rui Liu
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Angles in Hilbert's axioms for geometry

In Hilbert's axioms for geometry, the following elements are presented as undefined (meaning "to be defined in a specific model"): point, line, incidence, betweenness, congruence. In the euclidean plane $\mathbb{E}^2$ , for example, we define a…
rmdmc89
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