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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?

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3 is what you mean by no coordinate system. I don't really understand the question. – AnonymousCoward Dec 24 '11 at 18:19
the a priori status of geometry makes rigor more difficult. take a look at hilbert's axioms as opposed to euclid's. – yoyo Dec 24 '11 at 18:47
GottfriedLeibniz, is synthetic geometry independent of ZFC? It was my impression that it is assumed that points and lines are assumed to be mathematical objects drawn from ZFC. – user18063 Dec 24 '11 at 19:26
What does ZFC have to do with that? – Asaf Karagila Dec 24 '11 at 20:37
You can think of points and lines in euclidean geometry as axiomatic objects like euclid did: Or you can consider them as objects that result from ZFC, I don't see how this has much of an impact on doing geometry once you've decided you have points and lines though. – AnonymousCoward Dec 24 '11 at 20:42
up vote 18 down vote accepted

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

Given a circle with center $C$, and a point $A$ such that $|CA|$ is less than the radius of the circle. Then any straight line through $A$ intersects the circle.

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.

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Thank you for your erudite answer, Mr. Makholm. So, although, one can do geometry outside of ZFC, am I to understand that this is not the standard practice? And if one can do geometry outside of set theory, does that mean that, if one chooses this option, that the Cantor-Dedekind axiom is a sort of metamathematical axiom connecting two separate axiomatic systems? – user18063 Dec 25 '11 at 1:12
@user18063: True, it is not usually done this way. Most mathematicians consider set-theoretic and algebraic methods much easier and more powerful to work with than synthetic geometry, so there is little interest in doing real work the Euclidean way. – Henning Makholm Dec 25 '11 at 1:22
@user18063: You're probably right that the "Cantor-Dedekind axiom" is best characterized as a metamathematical tenet about the "intended models" of geometry and set theory, respectively. I think it ought to be provable as a model-theoretic property of Hilbert's axiom system. If, on the other hand, one views it as a statement about intuitive plane geometry, it provides an intuitive argument that Hilbert chose "the right way" to complete Euclid's notions. – Henning Makholm Dec 25 '11 at 1:25
+1. This might be slightly unrelated, but does Hilbert's axioms work only for three-dimensional Euclidean spaces, or can it extend to $\mathbf R^n$? Has anyone studied such higher (even arbitrary) dimensional generalisation of synthetic geometry? // There's a minor typo ("thises") in bullet point 4, 6th line. – Srivatsan Dec 25 '11 at 2:16
@Srivatsan: Wikipedia claims that "suitable changes" to Tarski's dimension-setting axioms will yield $n$-dimensional Euclidean space. In contrast, Hilbert's system had a primitive sort for each dimension and it is not clear to me how smoothly it extends to higher dimensions. – Henning Makholm Dec 27 '11 at 22:56

High school geometry, at its most rigorous, works off of some set of axioms for how points, lines, lengths, etc. should behave. These axioms are modeled by the behavior of points, lines, lengths, etc. in the Euclidean spaces $\mathbb{R}^n$, in the same way that specific groups such as the symmetric groups $S_n$ model the axioms for a group. In other words, the difference between the two points of view is, roughly speaking, the difference between "syntax" and "semantics." Both are parts of ordinary mathematics; neither stands outside of it.

Some things are easier to do syntactically, while others are easier to do semantically. From the perspective of modern mathematics, the major benefit of the semantic perspective on Euclidean geometry is that it allows you to talk about things that go beyond what the traditional axioms are allowed to talk about - things like the Euclidean group and so forth - and is more amenable to interesting and explicit generalizations.

For example, syntactically it's hard to decide whether non-Euclidean geometry makes sense. Semantically, however, it suffices to construct a model of non-Euclidean geometry (e.g. construct hyperbolic space).

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I see. So would you say there is no more to the intuitive picture of geometry than just that: a picture? I always thought that picture had more to it, a real ontological status. I guess not? – user18063 Dec 24 '11 at 20:07
@user18063: I don't know what you mean by that. – Qiaochu Yuan Dec 24 '11 at 21:08
@ Qiaochu Yuan: it seems to me that to the ancient Greeks, and to Euclid in particular, geometric figures really do exist and they look like they are pictured in diagrams. Therefore, in his Elements, Euclid sees himself as speaking about objects which one can 'see'. But, if I understand correctly, mathematicians today do not subscribe to this point of view: rather, to them, the picture we have of geometry is purely an informal one and meant only as an aid. Points and lines do not really look like anything, they're just tuples of real numbers and particular sets of these. Is that right? – user18063 Dec 24 '11 at 22:48
@user18063: points and lines are things satisfying certain axioms. To really help absorb this point it helps to strip away a few additional axioms and look only at projective, rather than Euclidean, geometry: then there are models in which "space" has only finitely many points ( which don't look very much like ordinary Euclidean geometry, but which nevertheless satisfy all of the axioms of projective geometry. – Qiaochu Yuan Dec 24 '11 at 23:41
@user18063: You seem to be thinking about Platonism. Most modern mathematicians are still Platonists, but actually doing mathematics requires no ontological commitments. – Zhen Lin Dec 25 '11 at 1:04

What is taught in high school and elementary school under the label geometry is somewhat of a societal decision. Recently the NCTM (National Council of Teachers of Mathematics) the professional organization for K-12 published a yearbook devoted to geometry which discusses emerging ideas about what is a good choice of content for K-12 geometry. The book is entitled Understanding Geometry for a Changing World (71st yearbook-2009) - Editors Tim Craine and Rheta Rubenstein.

About 20 years ago there was small conference of "research geometers" to discuss ideas for the survey course in geometry for college and how to interface high school and college geometry that COMAP sponsored. The "proceedings" (which I edited) appeared under the title: Geometry's Future (1991). The "book" is out of print but there are copies available in libraries and probably used copies are available. Some of the people whose essays appear are Bill Thurston, Tom Banchoff, Vic Klee, and Marjorie Senechal. There is also this recent item by Carl Lee:

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Interesting point. – user18063 Dec 26 '11 at 15:56

I encourage you to read this article (link below) by Marvin Jay Greenberg. The two related textbooks are the fourth edition of his Euclidean and Non-Euclidean Geometries and Robin Hartshorne's Geometry: Euclid and Beyond. In short, Hilbert's program was completed by Bachmann and Pejas. One may follow an entirely synthetic approach.

You can download the article itself from MARVIN ... Greenberg won an award for the article, and he is certainly concerned with questions of foundations. In particular, with some care, geometry can be considered entirely without recourse to numbers. Segment arithmetic includes ideas such as line segments being the same length, or one of them longer, but does not require assigning any number to a length. Plus, of course, I am in the article. I helped in writing a little part.

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Thank you everybody for your answers. Here's what I've gathered, feel free to criticize:

For sure, one can do geometry outside of set theory. Hilbert does not presume a background set theory when stating his axioms in his Grundlagen der Geometrie. This said, one can certainly find models of Euclidean geometry within set theory: $\mathbb{R}^2$ and $\mathbb{R}^3$ are the prime examples. Presumably because mathematicians today prefer to work with the full power of set theory, they actually define their Euclidean plane and space as $\mathbb{R}^2$ and $\mathbb{R}^3$. One could say that mathematicians choose to focus on a particular model of geometry, or at most those models that are built inside of set theory. This in fact poses no harm because Euclidean geometry, in Hilbert's form, is categorical: a same statement has the same truth value across all models of Euclidean geometry.

My final point, which is more tenuous and philosophical, concerns the status of the 'intuitive' picture of geometry. I'd like to think we can think of it as yet another model of geometry, one that lives inside the mind, abstracted from the world around us (something similar to what Kant would have to say).

Edit: To define circles, for example, it seems set theoretic language is inevitable. Hilbert says:

I make use particularly of Cantor’s theory of assemblages of points [...] If in our geometry we define a true circle as the totality of those points which arise by rotating about [a point] $M$ a point different from $M$ [...]

So apparently a 'set of points' is also an undefined term in Hilbert's treatment of geometry. I suppose that prior to the scandal that was Russel's paradox, mathematicians did not feel the need to spell out their assumptions about sets explicitly and took them for granted?

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(Part 1): Relatively few people who get a doctorate degree in mathematics would, if you asked what part of mathematics they work in, describe themselves as geometers. However, among those who call themselves geometers, almost none of them work on questions related to axiomatics or "foundations" any longer. Geometry has many parts: – Joseph Malkevitch Dec 26 '11 at 17:11
(Part 2): So people who do research in geometry often are attacking questions in one of these areas. Unlike algebra and analysis often the insights into important geometrical questions do not always provide tools for other "big" geometrical questions - resulting in the "accusation" that geometry consists of a bag of tricks. But, for those who love geometrical ideas this appears to be the nature of the "beast." – Joseph Malkevitch Dec 26 '11 at 17:11
(Part 3): A recent book that shows the range of our ignorance about "simple" geometrical phenomena is Geometric Folding Algorithms: Linkages, Origami, and Polyhedra by E. Demaine and J. O'Rourke (Cambridge U. Press, 2007). Symbolic algebra systems and other software systems rich tools for experimentation in many parts of mathematics. While many deep ideas are born of geometrical considerations the different parts of geometry often "mature" when they become more "algebratized." It is not a coincidence that synthetic Euclidean and projective geometry came before their analytical versions. – Joseph Malkevitch Dec 26 '11 at 17:14

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