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I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own sets of axioms.

I have two related questions:

1) What is the modern axiomatization of plane geometry? For example, when mathematicians speak of a point, a line, or a triangle, what does this mean formally?

My guess would be that one could simply put everything in terms of coordinates in R^2, but then it seems to be hard to carry out usual similarity and congruence arguments. For example, the proof of SAS congruence would be quite messy. Euclid's arguments are all "synthetic", and it seems hard to carry such arguments out in an analytic framework.

2) What problems exist with Euclid's elements? Why are the axioms unsatisfactory? Where does Euclid commit errors in his reasoning? I've read that the logical gaps in the Elements are so large one could drive a truck through them, but I cannot see such gaps myself.

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    $\begingroup$ On thing I've found eye opening is to consider the case where the "points" are pairs of rational numbers and the lines are usual lines with rational slope and rational y-intercept. This satisfies all of Euclid's axioms, but then, for example, the line $y = x + 1$ doesn't intersect the circle $x^2 + y^2 = 4$, even though it goes from "outside" to "inside" then back to "outside". So a simple statement like "A line segment which starts inside a circle and ends outside of it must intersect the circle" can't be proven in Euclidean geometry. $\endgroup$ – Jason DeVito Nov 10 '11 at 20:18
  • $\begingroup$ On the other hand, such constructions work as usual when both coordinates are in the "constructible numbers," call them $E,$ the smallest subfield of $\mathbf R$ such that, if $x \in E$ and $x > 0,$ then $\sqrt x \in E.$ $\endgroup$ – Will Jagy Nov 10 '11 at 22:40
  • $\begingroup$ @JasonDeVito, nice! So what minimal "missing" axiom is required to show that the line intersects the circle? $\endgroup$ – Paul Draper Jan 24 '15 at 18:11
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    $\begingroup$ @Paul: It's literally been years since I've thought about this. If I recall correctly, Hilbert's axioms fix everything up. I'm not sure which "minimal" axiom of Hilbert's would fix this particular issue, though. $\endgroup$ – Jason DeVito Jan 25 '15 at 1:14
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    $\begingroup$ For 2), one way is to read Euclid and look for such gaps, and when you find one, add in the needed axiom. Doing this, I think one comes away with the feeling that the gaps are actually not so many, and almost implicitly clear already in Euclid as to how to fix them, at least with our enormous hindsight. To me, there are basically 4 of them: the line through 2 pts is unique (Post.1), circles that pass inside and outside each other should meet (Prop.1), lines should separate the plane (Post.4), and rigid motions should be possible (Prop.4). Hartshorne makes all this very clear and precise. $\endgroup$ – roy smith Apr 22 '15 at 17:05
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I can recommend an article Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg, The American Mathematical Monthly, Volume 117, Number 3, March 2010, pages 198-219. One of the great strengths of the article is that I am in it. Marvin promotes what he calls Aristotle's axiom, which rules out planes over arbitrary non-Archimedean fields without leaving the synthetic framework. If you email me I can send you a pdf.

EDIT: Alright, Marvin won an award for the article, which can be downloaded from the award announcement page GREENBERG. The award page, by itself, gives a pretty good response to the original question about the status of Euclid in the modern world.

As far as book length, there are the fourth edition of Marvin's book, Euclidean and Non-Euclidean Geometries, also Geometry: Euclid and Beyond by Robin Hartshorne. Hartshorne, in particular, takes a synthetic approach throughout, has a separate index showing where each proposition of Euclid appears, and so on.

Hilbert's book is available in English, Foundations of Geometry. He laid out a system but left it to others to fill in the details, notably Bachmann and Pejas. The high point of Hilbert is the "field of ends" in non-Euclidean geometry, wherein a hyperbolic plane gives rise to an ordered field $F$ defined purely by the axioms, and in turn the plane is isomorphic to, say, a Poincare disk model or upper half plane model in $F^{\; 2}.$ Perhaps this will be persuasive: from Hartshorne,

Recall that an end is an equivalence class of limiting parallel rays

Addition and multiplication of ends are defined entirely by geometric constructions; no animals are harmed and no numbers are used. In what amounts to an upper half plane model, what becomes the horizontal axis is isomorphic to the field of ends. This accords with our experience in the ordinary upper half plane, where geodesics are either vertical lines or semicircles with center on the horizontal axis. In particular, infinitely many geodesics "meet" at any given point on the horizontal axis.

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  • $\begingroup$ That's new. I actually had to put in spacing between the $F$ and the exponent 2, otherwise unreadable. So it is F^{\; 2} $\endgroup$ – Will Jagy Nov 10 '11 at 21:11
  • $\begingroup$ That's gorgeous. $\endgroup$ – Robert Haraway Nov 10 '11 at 23:39
  • $\begingroup$ Just want to add that there is also good axiomatic material in Introduction To Geometry by Coxeter in the chapter Ordered Geometry.( e.g.If 3 points A,B,C on a sphere lie on a great circle you can't say that B is not between A and C, so for the Euclidean plane some foundation for "betweenness" on a line is needed.) $\endgroup$ – DanielWainfleet Oct 13 '15 at 3:11
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I know this is an old Q, but I'll post another answer anyway -- hopefully someone finds this useful.

Cool Qs. Anyway, for point 1), I don't think there is “the”, as in a single, such axiomatization. You mention Hilbert's axioms, that is one such axiomatization, and probably the one closest in “spirit” to Euclid's. Tarski's axioms are another, and have the interesting property that they can all be done in first-order logic (i.e. no set theory), but are much more abstract – e.g. we don't even have a concept of “line” as an object, everything is done in terms of points. Hilbert's system gives points, lines, and planes as the basic building blocks of the geometry. But both are just as good. There's also Birkhoff's axioms, which come closest to the idea of “doing it all with coordinates” in that they explicitly introduce the notion of real numbers into the geometry (whereas the former two simply have “continuity axioms” that, among other things, ensure that the geometry has the same amount of points on a line as there are real numbers).

It is also true that in modern maths, “Euclidean space” is defined as $\mathbb{R}^n$, but this is not usually considered an “axiomatic” approach, so does not fit your question. Rather, that is the “analytic” approach. In this approach, points are defined as ordered tuples of reals, lines are sets of such tuples satisfying relations, and triangles are then just the total collection of points on the lines between the 3 points defining the triangle. In the “axiomatic” or “synthetic” approach, points and lines are considered as undefined concepts – they have no formal mathematical definition, but instead we specify their behavior by axioms. Euclid didn't quite get this far (but of course, this was ancient maths we are talking about, so standards and what not were different then) – he attempted to “define” them, such as saying “a point is that which has no part”, “a line is breadthless length”, etc. which are themselves not mathematically useful – only intuitively useful – “definitions”. So in modern axiomatizations, at least one of these concepts is taken as primitive (in Hilbert's axioms, as mentioned, points and lines are both primitive, in Tarski's, there are only points, and indeed, since Tarski's theory does not involve sets, then one cannot even define a line by a set of points!), with no mathematical definition.

Point 2) is also interesting. There are indeed, many problems. One such set of problems can be seen by the very first proposition of the Elements. This is the construction of an equilateral triangle on a segment. Euclid says, you can construct two circles, centered at each point of the segment, with radii equal to the given segment. Then the point of intersection of the circles, plus the two points at the ends of the segment, define the triangle, and it is equilateral. The trouble with his construction is that there are insufficient axioms to justify all of it. That the line segment and the circles exist is, of course, provable, but he runs into trouble when he mentions the points of intersection. There are no postulates in Euclid's axioms that deal with intersections except for the parallel postulate! This is a horrific omission on his part – certainly one that I'd say would leave a gap “big enough to drive a truck through”, since it essentially renders unjustified essentially all his compass-and-straightedge constructions (at least, any time he needs to take an intersection of lines and circles, he can't!). Another problem with the proof, and this was pointed out all the way back in ancient Greece by Zeno of Sidon (not the same Zeno of Zeno's Paradoxes fame – that was Zeno of Elea), is that there is no way to know that when we generate the lines from the segment tips to the circle intersection point, that they even form an equilateral triangle and don't meet “ahead of time”. What would have been needed would have been a postulate on the uniqueness of a straight line between two points. Hilbert's axioms address these challenges: he gives an axiom that says that the straight line between two points is uniquely determined by two given points. He also gives the “continuity axioms”, which guarantee the circles and lines will intersect (and his continuity axioms are even stronger – they guarantee the existence of all real numbers in the plane, when you only need the “surd numbers” as they're called to get the circle intersections right.).

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  • $\begingroup$ IIRC, Hilbert didn't have lines any more than Tarski did; Hilbert presumes a set theory and abstractly asserts certain sets of points are to be called lines. In Tarski's, it's easy enough to define a line to be a pair of distinct points, modulo a relation that says when two pairs define the same line. And either way, it's easy enough to mechanically translate the axiomatization to one where lines are among the fundamental notions. $\endgroup$ – Hurkyl Mar 31 '13 at 20:39
  • $\begingroup$ Actually, Hilbert's axioms do include the notion of "straight line" as a primitive notion. It's things like like segments, and circles, that are defined as sets of points. $\endgroup$ – The_Sympathizer Apr 1 '13 at 22:11
  • $\begingroup$ Hrm. I suppose the presentation I was recalling decided to "streamline" things then. $\endgroup$ – Hurkyl Apr 1 '13 at 22:21
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    $\begingroup$ The key thing to remember about Euclid's postulates is that they are all satisfied by the "plane" $\mathbb{Q}\times\mathbb{Q}$. This is another way to see why they have so many problems proving that points of intersection exist. $\endgroup$ – Carl Mummert Feb 22 '15 at 1:06
  • $\begingroup$ @mike4ty4 Great answer! I'm still a little confused. Maybe you can help me. :) Question (1)Are Hilbert's Axioms rigorous enough(i.e. all correct)? (2)You said that, "in modern maths, 'Euclidean space' is defined as $\Bbb R^n$." So does it mean, in modern math, points, lines, triangles, etc, are defined to be some particular elements or subsets of $\Bbb R^n$? (In this way, it seemingly means we don't use the Hilbert's axiom anymore; they are not useful.) $\endgroup$ – Eric Mar 24 '17 at 18:21
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For the first question, when the Euclidean plane appears in modern mathematics, it is almost universally in the guise of $\mathbb R^2$ with the "Euclidean norm" $\|(x,y)\|=\sqrt{x^2+y^2}$.

Congruence has a nice representation in terms of isometric (i.e. length-preserving) transformations of $\mathbb R^2$, which turn out to be everything that can be expressed using translation (addition of a constant vector) and orthogonal linear transformations. These concepts are slightly less immediate than Euclid's intuitive motions of the plane, but are immeasurably more productive in terms of generalizing to other more abstract settings than the plane. Because of these generalizations, one can use one's geometric intuition about things that we would otherwise have no easy intuitive way to think about -- from a modern viewpoint this is a major triumph of the vector approach over synthetic geometry.

Similarly(!), similarity can be understood as the combination of isometric transformations and linear scalings of the plane.

Most mathematicians would say that the gains of generalization by far outweigh the minor technical hassle of proving elementary geometric facts by algebraic means. This doesn't mean that the classical proofs are completely abandoned. A basic fragment of synthetic geometry continues to be relevant as proof that everyday geometric intuition applies to $\mathbb R^2$ at all.

For axioms, see vector space and normed vector space. (These are already more general than the plane -- for the plane itself, identified as $\mathbb R^2$, we don't need any new axioms but the ones that define $\mathbb R$ and the definitions of the various vector operations).

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My currently-preferred definition is:

Definition. $\mathbb{E}_2$ is the unique Euclidean space (up to isomorphism) whose underlying translation space is $2$-dimensional.

Whether or not this is actually an axiomatization is debatable; certainly, it is not first-order, and it cannot be implemented without a background set theory. I tend to view it as a mere definition, for that reason.

If you want to see Euclidean geometry developed rigorously from this perspective, check out Audin's Geometry. By the way, this is also the definition given at nLab. A more sophisticated approach is to define $\mathbb{E}_2$ as a particular real Riemannian manifold. This is probably closest to how modern geometers think about Euclidean geometry, but it misses important simplifications that only make sense in the flat, Euclidean case.

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    $\begingroup$ I mean, "a 2-dimensional Euclidean space" is an axiomatization (though a good deal of the important axioms are concealed in an axiomatization of $\mathbb{R}$ itself). $\endgroup$ – Eric Wofsey Oct 13 '15 at 2:27
  • $\begingroup$ @EricWofsey, true. I've edited appropriately. $\endgroup$ – goblin Oct 13 '15 at 4:15
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For example, the proof of SAS congruence would be quite messy.

I don't understand what proof you have in mind. Let $p_1, p_1 + v_1, p_1 + v_2$ and $p_2, p_2 + u_1, p_2 + u_2$ be two triangles in the plane (where $p_i$ are points and $v_i$ are vectors) such that $|v_i| = |u_i|$ and such that the angle between $v_1$ and $v_2$ is equal to the angle between $u_1$ and $u_2$. We want to show that there exists an isometry of the plane sending one triangle to the other.

By translation (which is clearly an isometry) we may assume WLOG that $p_1 = p_2$. By rotation (again clearly an isometry) we may assume WLOG that $v_1 = u_1$. So the problem reduces to showing that $v_2$ is uniquely determined by its length and its angle to $u_1$. But this is just the statement that polar coordinates are unique (away from the origin), which can be proven in any number of ways in this framework and is a useful and important fact in its own right.

In any case, Henning is right:

Most mathematicians would say that the gains of generalization by far outweigh the minor technical hassle of proving elementary geometric facts by algebraic means.

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Euclid did not have the concept of real number and Hilbert deliberately avoided it in his axiom system. I believe some modern systems that are otherwise similar to those ("synthetic" geometry) take that concept for granted and go on from there.

However, just to add an item to the list of systems: the Huzita–Hatori axioms, although stated as axioms about paper folding, may amount to a system of axioms of plane geometry. The Wikipedia article states: "compass and straightedge geometry solves second-degree equations, while origami geometry, or origametry, can solve third-degree equations, and solve problems such as angle trisection and doubling of the cube."

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The following axiomatization of the Euclidean plane can be found starting on p. 168 of Elementary Geometry by Agricola and Friedrich:

A geometric plane consists of

  • a set $\mathcal{P}$, whose elements we call points,
  • a set $\mathcal{Z} \subset \mathcal{P}\times\mathcal{P}\times\mathcal{P}$. If the triple $(A,B,C)$ of points in $\mathcal{P}$ lies in $\mathcal{Z}$, we shall say that the point $B$ lies between the points $A$ and $C$.

Such a plane satisfies the following axioms:

1. There exist three distinct points $A_0, B_0, C_0 \in \mathcal{P}$, which do not belong to the betweenness relation $\mathcal{Z}$ in any order, in other words $$(A_0, B_0, C_0) \not\in \mathcal{Z}, \quad(A_0, C_0, B_0) \not\in \mathcal{Z},\quad (B_0, A_0, C_0) \not \in\mathcal{Z}, \quad (B_0, C_0, A_0)\not\in\mathcal{Z}, \quad(C_0, A_0, B_0)\not\in\mathcal{Z},\quad (C_0, B_0, A_0)\not\in \mathcal{Z}$$

2. If one of the points $A,B,C$ lies between the others, then the three points are different.

3. Given two distinct points $A$ and $B$ there exists a point $C$ such that $B$ lies between $A$ and $C$.

4. If $B$ lies between $A$ and $C$, then $B$ lies between $C$ and $A$.

5. If $A,B,C$ are three points in $\mathcal{P}$, then at most one of these points lies between the other two.

6. If one of the points $A,B,C$ lies between the other two and one of the points $A,B,D$ also lies between the other two then one of the points $B,C,D$ likewise lies between the other two.

Let $A,B \in \mathcal{P}$ be two distinct points. The segment $AB$ consists of $A,B$ and all points lying between $A$ and $B$.

Let $A$ and $B$ be two distinct points. The line $\mathcal{L}(A,B)$ consists of $A$, $B$, and all points $C$ with the property that one of the points $A,B,C$ lies between the other two.

Two lines $\mathcal{L}$ and $\mathcal{L'}$ are called parallel if either they are the same or they have an empty intersection $\mathcal{L}\cap\mathcal{L'}=\emptyset$.

7. If $A,B,C$ are three distinct points not in any betweenness relation, $S$ is a point of the segment $AB$, and $T$ is a point of the line $\mathcal{L}(A,C)$ which does not lie in the segment $AC$, then the line $\mathcal{L}(S,T)$ contains at least one point of the segment $BC$.

8. If $A, B,C$ are three distinct points, not in any betweenness relation, then the union $$\mathcal{P}=\bigcup_{S \in BC} \mathcal{L}(A,S) \cup \bigcup_{T \in AC} \mathcal{L}(B,T) \cup \bigcup_{U \in AB} \mathcal{L}(C,U) $$ of all lines which contain a vertex of the triangle $\Delta (A,B,C)$ and a point of the respective opposite side is equal to the whole plane $\mathcal{P}$.

We postulate a new basic object $$d: \mathcal{P}\times\mathcal{P} \to [0,\infty). $$ The number $d(A,B)$ is called the distance from the point $A$ to the point $B$. The distance $d(A,B)=0$ vanishes precisely when the points $A$ and $B$ are equal. The distance is, further, to be symmetric, $d(A,B)=d(B,A)$, and to satisfy the triangle inequality $$d(A,B) \le d(A,C) + d(C,B). $$

9. The point $C$ lies in the segment $AB$ if and only if $$d(A,B)=d(A,C)+d(C,B). $$

10. The pair $(\mathcal{P},d)$ is a complete metric space; i.e. each Cauchy sequence in $\mathcal{P}$ converges to a point in $\mathcal{P}$.

11. Let $\mathcal{L}$ and $\mathcal{L'}$ be two lines that cut each other at a point $P$. Further, let $A, A_1 \in \mathcal{L}$ and $B, B_1 \in \mathcal{L'}$ be two points on these lines such that $P$ lies between $A$ and $A_1$ on $\mathcal{L}$ and between $B$ and $B_1$ on $\mathcal{L'}$. If $d(P,A)=d(P,A_1)$ and $d(P,B)=d(P,B_1)$, then $d(A,B)=d(A_1,B_1)$.

12. If $A,B,C$ are three distinct points not in any betweenness relation, and if $A_1,B_1$ are two other points with $d(A,B)=d(A_1,B_1)$, then there exists a point $C_1$ with $d(A,C)=d(A_1,C_1)$ and $d(B,C)=d(B_1,C_1)$.

An isometry of the geometric plane $\mathcal{P}$ is a bijective mapping $f: \mathcal{P} \to \mathcal{P}$ that preserves distances; i.e. $d(f(A),f(B))=d(A,B)$. Two subsets of $\mathcal{P}$ are called congruent if there is an isometry that maps these sets bijectively one onto the other.

13. If $A,B,C$and $A_1,B_1, C_1$ are sets of three points of $\mathcal{P}$ and if $$d(A,B)=d(A_1,B_1), \quad d(A,C)=d(A_1,C_1), \quad d(B,C)=d(B_1,C_1), $$ then there is an isometry $f$ with $f(A)=A_1, f(B)=B_1$, and $f(C)=C_1$.

A Euclidean plane is a geometric plane satisfying this following axiom, called the parallel postulate:

14. For each line $\mathcal{L}$ and each point $A$ not lying on it there exists a unique line parallel to $\mathcal{L}$ through the point $A$.

Note that the existence of such a line follows from the first 13 axioms, but the uniqueness of the line must be an additional axiom -- for instance hyperbolic geometry satisfies the first 13 axioms, but it does not satisfy the parallel postulate. The first 13 axioms have to be modified somewhat for non-Euclidean geometries (e.g. spherical geometry) where the existence of a parallel line is not guaranteed.

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Any axiomitization of 2-dimensional Euclieadn geometry is unjustifiable. In such a system, you just assume without proof that the undefined concept of distance satisfies certain intuitive properties and prove other properties from it like the Pythagorean theorem. If you invent a space and explicitly define a lot of the relations on it like distance and show that those definitions satisfy the intuitive properties of 2-dimensional Euclidean geometry, it follows that that space can be represented by R^2 and that for any points (x1, y1) and (x2, y2), the distance from (x1, y1) to (x2, y2) is sqrt((x2 - x1)^2 + (y2 - y1)^2). We can define the distance formula to be sqrt((x2 - x1)^2 + (y2 - y1)^2) without giving a reason then show that it satisfies the intuitive properties of distance. After we show that it satisfies the intuitive properties, we can derive from that fact that that the formula is sqrt((x2 - x1)^2 + (y2 - y1)^2) but there's no need because that can be proven directly from the definition of distance. Some people might want to know why it was defined that way so for them, it might suffice to show that that formula is the unique formula that satisfies the intuitive properties of distance. Those people might also be satisfied with seeing the distance formula being proven by defining it them showing that it satisfies those intuitive properties then reproving it from those properties because from that, they can figure out how to prove that that formula is the unique formula that satisfies the intuitive properties of distance. The article http://speedydeletion.wikia.com/wiki/Distance that I wrote actually gives an incomplete proof that that formula is the unique formula that satisfies the intuitive properties of distance. A complete proof probably includes constructing the set of all real numbers with operations from the power set of the set of all natural numbers in Zermelo-Fraenkel set theory then showing that that set is a totally ordered field, similar to what I described in my answer at What is a natural number?.

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