Talk:Birkhoff's axioms

Postulate 2: “There is one and only line...” - Do we have a missing ‘one’ here?

Missing Context for Birkhoff’s Axioms / Clarity Issues

This article lacks clarity because of the following six issues:

1. It fails to specify the abstract system to which Birkhoff’s axioms apply.
2. It fails to explain what the function $d$ is.
3. It fails to explain what the function $\angle$ is.
4. It fails to define the term ‘triangle’.
5. It fails to define the term ‘ray’.
6. It fails to explain precisely what the continuity condition in Postulate III means.

One way to resolve the first three issues is to first posit the existence of a quadruple $({\mathcal {P}},{\mathcal {L}},d,\angle )$, where:

• ${\mathcal {P}}$ is a set of abstract objects, whose elements we call points;
• ${\mathcal {L}}$ is a set of subsets of ${\mathcal {P}}$, whose elements we call lines;
• $d$ is a function from ${\mathcal {P}}\times {\mathcal {P}}$ to $\mathbb {R} _{\geq 0}$, which we call a distance function;
• $\angle$ is a function from $\{(A,O,B)\in {\mathcal {P}}\times {\mathcal {P}}\times {\mathcal {P}}\mid A,B\neq O\}$ to $\mathbb {R} /2\pi \mathbb {Z}$, which we call an angle measure.

Birkhoff’s axioms are then applied to the quadruple $({\mathcal {P}},{\mathcal {L}},d,\angle )$, which serves as an abstract system, so as to imbue it with desirable geometrical properties.

Now, a triangle should be defined as a triple of distinct points.

Next, a ray through a point $O$ should be defined as a set of points of the form $\{A\in l\mid f(A)\geq f(O)\}$, where $l$ is a line containing $O$, and $f$ is a bijection from $l$ to $\mathbb {R}$ that satisfies Postulate I.

Finally, the continuity condition in Postulate III should be explained as follows. Let $l$ be a line that does not contain $O$, and let $f$ be a bijection from $l$ to $\mathbb {R}$ that satisfies Postulate I. Define a function $R$ from $l$ to the set of rays through $O$ such that for every $B\in l$, the ray $R(B)$ through $O$ is the unique one that contains $B$. If $\alpha$ is a bijection from the set of rays through $O$ to $\mathbb {R} /2\pi \mathbb {Z}$ that satisfies the first half of Postulate III, then the function

$\left\{{\begin{matrix}\mathbb {R} &\to &\mathbb {R} /2\pi \mathbb {Z} \\x&\mapsto &\alpha \!\left(R\!\left({f^{-1}}(x)\right)\right)\end{matrix}}\right\}$

is required to be continuous.

Leonard Huang (talk) 21:22, 29 November 2016 (UTC)

While I can not fault your above analysis, I do question its suitability in this article. Birkhoff presented his ideas in the original paper as a kind of "proof of concept" indicating how a simple set of axioms based on the physical experience of measurement could be used to develop Euclidean geometry. He expressed surprise that so many people took this as a serious attempt to produce an alternative to Euclid's axioms. When these axioms have been used (even by Birkhoff in his geometry textbook) they have to be placed in context and provided with the appropriate definitions, much as you have done above. Since this article is about the axioms, this additional framework is not really a part of the article. In a different article, concerned with using Birkhoff's axioms to construct Euclidean geometry, this material would be appropriate. But even there, the material would have to be based on previously published work else it would be considered WP:OR.--Bill Cherowitzo (talk) 04:00, 30 November 2016 (UTC)

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