Okay, so we've all seen Euclidean geometry in primary and high school. Back then, I really thought of points as indivisible entities in space and lines as 'breadthless lengths'. As far as I could tell, so did the other students and the teachers. This is the kind of geometry I mean when I say 'high school geometry'. In contrast, in higher mathematics, we commonly define the Euclidean plane and space as $\mathbb{R}^2$ and $\mathbb{R}^3$. This begs the question: what is the status of high school geometry from the mathematician's perspective? Is it simply an informal picture, just like a drawing of a graph? Or is it something more, a mathematics all of its own, separate from ZFC set theory?
[Math] The status of high school geometry
geometryphilosophysoft-question
Related Solutions
Basically, to develop "formally" a geomety you have two ways; call them analytic and synthetic respectively.
Analytic
This is our "good old" Analytic geometry :
a point in the space is a ordered triple of real numbers : $(x_1,x_2,x_3)$
a line is the totality of points $(x_1,x_2,x_3)$ such that $u_1x_1 + u_2x_2 + u_3x_3 = 0$, where at least one $u_j (j = 1,2,3)$ is different from zero
and so on ...
But real numbers are definable in set theory; thus - in principle - you can translate into set-theoretic notation the equation of the line.
Synthetic
See Edwin Moise, Elementary Geometry from an Advanced Standpoint (3rd ed - 1990), page 43 :
space will be regarded as a set $S$; the points of space will be the elements of this set. We will also have given a collection of subsets of $S$, called lines, and another collection of subsets of $S$, called planes.
Thus the structure that we start with is a triplet : $<\mathcal S, \mathcal L, \Pi>$, where the elements of $\mathcal S, \mathcal L, \Pi$, and are called points, lines and planes, respectively.
Our postulates are going to be stated in terms of the sets $\mathcal S, \mathcal L$, and $\Pi$.
Here are the first two postulates :
I-0 : All lines and planes are sets of points.
I-1 : Given any two different points, there is exactly one line containing them [we can "trivially" express the fact that the point $Q$ is contained into the line $l$ with the formula : $Q \in \mathcal S \land l \in \mathcal L \rightarrow Q \in l$ ].
We write $\overline{PQ}$ for the unique line containing $P$ and $Q$.
We define the relation of betweenness between (sic !) three points $P, Q, R$.
Then [see pages 64-65] : if $R,Q$ are two points, the segment between $R$ and $Q$ is the set whose points are $R$ and $Q$, together with all points between $R$ and $Q$.
The ray $\overrightarrow {AB}$ is the set of all points $C$ of the line $\overline {AB}$ such that $A$ is not between $C$ and $B$. The point $A$ is called the end point of the ray $AB$.
An angle is the union of two rays which have the same end point, but do not lie on the same line. If the angle is the union of $\overrightarrow {AB}$ and $\overrightarrow {AC}$, then these rays are called the sides of the angle; the [common] end point $A$ is called the vertex.
Finally, you can "close the circle" between this two approaches.
Assuming that we have defined the set $\mathbb N$ of natural numbers inside set theory [but I prefer to say that we have defined a model of the natural number system], and then the set $\mathbb R$ of real numbers, we can use $\mathbb R^3$ and call it : (three-dimensional) space.
Comment
What have we gained so far ? I think nothing more and nothing less than what we already have with Descartes' discovery of analytic geometry : an "embedding" of the euclidean geometry into the "cartesian plane".
Of course, the "basic" set-theoretic language gives us a powerful tool for expressing also geometrical "facts" : we can write $P \in l$ for : "the point $P$ is contained into line $l$", we can write $l_1 \cap l_2 \ne \emptyset$ for "two lines intersect each other", ...
But I think that speaking of "foundation for most of modern mathematics" can be mesleading.
Geometry is really a very broad term and it encompases many different realms of mathematics.
Usually, the main dichotomy is between algebraic geometry and differential geometry.
For algebraic geometry, 2 series come to mind:
1.Kenji Ueno, Algebraic Geometry (these are 3 monographs by AMS) 2.Cox, O'Shea , Little: 1)Ideals, Varieties, and Algorithms/2) Using Algebraic Geometry
Also there are the vast notes by Ravi Vakil on his website, which I highly recommend (I think that studying Ueno and then Vakil will give you a very good foundation in algebraic geometry).
Now for Differential Geometry again 2 series come to mind:
- John Lee: Topological Manifolds/Smooth Manifolds/Curvature
- Tu: An Introduction to Manifolds/ Differential Forms in Algebraic Topology (allong with Bott)/Connections, Curvature and Characteristic Classes.
Also, one could start reading Guillemin & Pollack: Differential Topology and then the classical Milnor: Topology from the differential viewpoint.
Finally, you could also take a look in Berger: A panoramic view of Riemannian Geometry where the author aims tou give a quick but to the point description of all the areas of that vast subject.
Best Answer
One can do Euclidean geometry as a completely formal game of symbolic logic. Euclid's axioms are almost sufficient for this, except that they lack formal support for some "obvious" continuity properties such as
David Hilbert, one of the pioneers of the formalist viewpoint, developed an axiomatic system that closes these gaps and allows any Euclidean theorem about a finite number of lines, circles, and points to be proved completely formally. One can work in this system without any reference to arithmetic or set theory, considering Hilbert's geometry axioms as an alternative to, say, ZFC as one's formal basis for one's reasoning.
(Edit: Oops, my history was slightly wrong. Hilbert's axiomatic system was not completely formal. Alfred Tarski later developed a completely formal system; what I say here about Hilbert rightfully ought to read Tarski instead.)
Granted, when one does that one doesn't necessarily think of lines and points as "really existing" in some Platonic sense -- after all, the basic idea of formalism is that no mathematical objects "really exist" and it's all just symbols on the blackboard that we play a parlor game of formal proofs with. But that does not mean that it is necessary, or even desirable, to consider points to be coordinate tuples while playing the game. Indeed, many mathematicians would probably agree that the points and lines of Hilbert's geometric axioms exist "in and of themselves" to at least the same extent that the sets of ZFC (or, for example, the real numbers) exist "in and of themselves".
There are some additional points about this state of things that have the potential to cause confusion, but do not really change the basic facts:
When I speak of Hilbert's axioms as an "alternative" to ZFC, I don't mean that they can be used as a foundation for all of mathematics they way ZFC is -- because they have not been designed to fill that role. I mean merely that they occupy the same ontological position in terms of which concepts one needs to already be familiar with in order to work with them. Perhaps "parallel" might be a better word than "alternative".
The rules of what consists valid formal proofs (in ZFC or geometry) are ultimately defined informally. One may construct a formal model of the rules, but that just punts the informality to the next metalevel, because reasoning about the formal model itself needs some sort of foundation.
When one does formalize the rules of formal proofs, one often does that in a set-theoretic setting. However, this doesn't mean that a different theory such as Hilbert's geometry "depends on" set theory in a fundamental way. Remember that the set theory we use to formalize logic can itself be considered a formal theory, and at some point we have to stop and be satisfied with an informal notion of proof (it cannot be turtles all the way down). And there is no good reason why there has to be a set-theoretic metalayer below the geometry before we reach the inevitable point of no further formalization.
This does not mean that formalizing the rules of formal geometric proof is a pointless exercise. Doing so can tell us things about the axiomatic system that cannot be proved within the formal system itself. In particular, if we formalize the axiomatic system within set theory, we get access to the very strong machinery of model theory to prove facts about the axiomatic system. This is where numbers and coordinates enters the picture, as described in Qiaochu's answer. Using these techniques, one can prove [as a theorem of set theory] that every formal statement about a finite number of lines, points and circles can be either proved or disproved using Hilbert's axioms.
Hilbert's system does not allow one to speak about indeterminate numbers of points in a single statements -- so one cannot prove general theorems about, say, arbitrary polygons. This is by design. The kind of reasoning usually accepted for informal proofs about arbitrary polygons is so varied that formalizing it would probably just end up being a more cumbersome way to do set theory, and there's not much fun in that.