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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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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. –  Jason DeVito Nov 10 '11 at 20:18
    
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 $\sqrt x \in E.$ –  Will Jagy Nov 10 '11 at 22:40

5 Answers 5

up vote 13 down vote accepted

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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That's new. I actually had to put in spacing between the $F$ and the exponent 2, otherwise unreadable. So it is F^{\; 2} –  Will Jagy Nov 10 '11 at 21:11
    
+1 for 3 outstanding references. –  Mathemagician1234 Nov 10 '11 at 23:12
    
That's gorgeous. –  Robert Haraway Nov 10 '11 at 23:39

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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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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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. –  Hurkyl Mar 31 '13 at 20:39
    
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. –  mike4ty4 Apr 1 '13 at 22:11
    
Hrm. I suppose the presentation I was recalling decided to "streamline" things then. –  Hurkyl Apr 1 '13 at 22:21

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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