# Why Do The Axioms of Euclidean Geometry Not Need To Include the Definition of Space?

EDIT: update, I found that Euclid's axioms are not considered rigorous. David Hilbert did a full axiomatization of Euclidean Geometry (1899 in his book Grundlagen der Geometrie--tr. The Foundations of Geometry). To do so, he required 6 primitive terms which were undefined, including points, lines, and planes. Lines and planes are both spaces. Therefore, a true axiomatization of Euclidean Geometry does in fact require space.

I left the rest of the post as it was below, but I believe ^^this is the answer. ...........

aside: Here's idea: Euclidean geometry is done using points and lines (0 and 1 dimensional spaces embedded in 2 dimensional space.) Why stop there? Why not do geometry with plane, line, point constructions embedded in 3 space? Or 3-space constructions in 4-space? The intersection of two lines is a point, the intersection of two planes would be a line, and the intersection of two "non parallel" 3-spaces would be a plane.

Here are the axioms of Euclidean Geometry:

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

That's without even including the parallel postulate.

Here are the axioms: http://mathworld.wolfram.com/EuclidsPostulates.html

EDIT: here's a possible first postulate that I came up with. Timax's postulate: There exists a 2-dimensional space which has distance-preserving maps. (maybe some linear algebra or topology would be necessary to define it fully)

But how can you even say "points exist" without first saying "space exists"? If points are defined as being mathematically represented by co-ordinates, (eg (23, 1, 9) is a point in 3-space,) then it seems the first axiom would have to be "3-space exists", and "real numbers exist" in order to create co-ordinates.

It seems pretty obvious that space must come before points.

I guess you could say a point exists, but to get beyond a zero dimensional space, and to get more than 1 point, you would need to establish that space exists. You can't have 2 points without at least 1 dimension. To get more than 1 line, you have to have 2 dimensions, to get more than 1 plane, you have to have 3 dimensions.

This is probably why the concept of "space" always seems so mysterious, BECAUSE THE EXISTENCE OF SPACE IS AN AXIOM THAT WAS NEVER STATED!

Here's another idea: Do Euclid's Elements-style constructions in 3 dimensional space or even N dimensional space, instead of only 2 dimensional space. Has that been done?

2015 M. Wanzek

• You are equating points with coordinates. Geometry doesn't need coordinates, so its basic axioms don't use them... – Michael Biro Jan 23 '15 at 13:22
• But it still needs SPACE to put the points in. Where are the points going to exist if they don't exist in space? – timax Jan 23 '15 at 13:33
• "Space exists" is a terribly vague axiom. Euclid's axioms are concrete - in between any two points there exists a line segment, for instance. They are concrete in the sense that you can visualize precisely what they mean, and more importantly, when he invokes them in a proof, it's clear how exactly they're being invoked and why they're needed. He has two points A and B and needs to claim there is a line segment between them - so he needs that axiom. Can you give an example where Euclid is implicitly invoking "Space exists"? Without any handwaving? – Jack M Jan 23 '15 at 14:38
• What kind of new geometry can be done with assuming space as an axiom? The point of listing axioms, at least for a modern mathematician, is to generate the true statements from the axioms and rules of inference. If the axiom of space were imdependent and generated different results, then maybe you would be more pursuasive. Axioms are not there to say anything about the "real world" other than not be contradictory and plausibly complete in some sense. – Rachmaninoff Jan 23 '15 at 14:45
• @timax: It seems to me your primary objection/unease (and usage of the verb "exist") is philosophical, not mathematical. Separately, your post poses multiple questions covering a wide range. It's possible this site's format isn't a good match, unless you can focus your question...? Be that as it may: 1. Yes, $n$-dimensional Euclidean geometry has been studied; 2. It's (mathematically) possible to have two points without having a one-dimensional "ambient universe". – Andrew D. Hwang Jan 23 '15 at 15:22

## 7 Answers

Why would you have to say “space exist” before you can say “points exist”? In order to have something that contains all those points? But then by the same reasoning, you can't say “space exist” without first postulating the existence of some entity that can contain spaces. Before you know it, you require an infinite regress of existence statments.

In short, “points exist” is a fine start. It was good at Euclid's time, and it's good now.

• I disagree. You don't need an infinite number of existance statements. You just need to say that 3-space or 1-space, or 5-space, or N-space or whatever exists, and N needs to be enough to contain whatever geometry you are doing. – timax Jan 23 '15 at 13:37
• In Euclid's case, he neglected to first have a postulate that 2-space exists. – timax Jan 23 '15 at 13:46
• You still haven't presented a single argument (as far as I can see) explaining why you need such an existence statement in the first place. As an anology, consider the Peano axioms for the natural numbers. They make sense without any mention of the set of all natural numbers. In fact, some mathematicians will deny the existence of this set, despite agreeing that natural numbers exist. And in Zermelo–Fraenkel set theory, there really isn't any object containing all sets. – Harald Hanche-Olsen Jan 23 '15 at 14:45
• @timax The existence of 2-space is implied by the definitions and axioms of Euclidean geometry. Think of it this way: if the existence of space is required for the existence of points, then "points exist" implies that space exists. – KSmarts Jan 23 '15 at 21:15
• @Harald Hanche-Olsen I guess this may be the answer. However, maybe my issue is that I was under the assumption that axioms were supposed to be statements so obvious that they had to be accepted. However, with the axioms of Euclidean geometry, he definetly failed to first define 2 space. In 3-space, or 1-space his geometry wouldn't work. So it is in fact neccessary to specify that he is talking about 2-space. It wasn't until Decarte, that the coordinate system was invented, so its unlikely that Euclid knew about the coordinate system. – timax Jan 24 '15 at 1:05

Because Euclid axioms define a space.

• Could you add some explanation to this answer? – Trevor Wilson Jan 23 '15 at 19:55

A space is a collection of objects that obey certain properties. That is, a space is a set with a structure. In this sense, Euclid's geometry doesn't need to assume the existence of a space, because he defines Euclidean Space. Consider also that Peano doesn't assume the existence of numbers, nor do Zermelo and Fraenkel assume the existence of sets—because they define them.

Of course, Euclid did his work well before the foundational crisis of mathematics, so some of his work was not what we would consider rigorous. He defines a point simply as "that which has no part" and a line as "breadthless length". If this is what is bothering you, more rigorous definitions have since been given.

By defining objects, we specifiy (or at least imply) what set we are working with—the set of points, lines, circles, etc.. Since we can consider lines, circles, and other objects as sets of points with certain properties, we can just say that our universe is the set of points. Also, we can say that we are not working in $0$ or $1$ dimensions because the definitions and axioms refer to intersections of lines, so we must be working in a set where such things are possible. In modern terms, standard semantics of logic require that the domain of discourse is nonempty.

The structure of the space is then specified by Euclid's Postulates, that is, the axioms. As you are no doubt aware, there other geometries that have different axioms, and therefore different structures.

In Euclidean geometry, you may refer to points, lines, and circles; implicitly and conceptually, these all reside within some "space", but Euclid's geometry doesn't include the vocabulary to refer to this whole space. Modern axiomatic mathematics certainly does have the capability to refer to the full entity of Euclidean space, naively represented as $\mathbb R^n$, that is, the tuples or vectors such as $(23, 1, 9)$ that you mention, but this is not necessary to describe the objects within Euclidean space, which is what Euclid did.

Another point that I suppose is worth mentioning: in your proposed "novel idea" that you added to the beginning of the question, you say the intersection of two lines is a point, the intersection of two planes is a line, the intersection of two 3-spaces is a plane, etc. It's an easy trap to fall into to extrapolate patterns like this, but consider two lines in 3-dimensional space: the vast majority of pairs (meaning: those in "general position") do not intersect at all; the behavior of the intersection of two such spaces is dependent on the ambient space.

The general rule here is encapsulated in Transversality theory: when two subspaces in general position are intersected, their codimensions, the dimension of the ambient space minus the dimension of the subspace, add. So for example, the intersection of two 3-spaces in general position inside 4-dimensional space is $4-(4-3)-(4-3) = 2$-dimensional, and the intersection of two 3-spaces in general position inside 5-dimensional space is $5-(5-3)-(5-3)=1$-dimensional. Two planes in general position in $4$-dimensional space intersect at a point.

• So you are saying, he wanted to define space, but he didn't have the vocabulary to do it? I guess that would make sense, but now that the vocabulary IS available, it should be updated. Also, what do you think of the idea of doing Euclid's Elements type constructions in higher dimensions? Euclidean geometry is done using points and lines (0 and 1 dimensional spaces embedded in 2 dimensional space.) Why stop there? Why not do geometry with plane,line,point constructions embeded in 3 space? Or 3-space constructions in 4-space? – timax Jan 23 '15 at 14:24
• That misses my point: you don't need to describe an entire entity just to be able to describe the objects within the entity. Euclid's axioms have the advantage that they are both simple and powerful, hence to this day we still teach them to our children. More advanced geometry can be saved for students who are interested in studying that in higher education. And yes, everything you suggest has been done. – Dustan Levenstein Jan 23 '15 at 14:34
• They may be simple and powerful, but they aren't complete, because it presumes 2-dimensional space. And I think the most logical explaination for why he didn't define space, is that he didn't have the mathematical tools to do so. I maintain that it is absolutely necessary to define space first. 99% of euclidean geometry is due to the space itself and its properties. Points and lines would only have their characteristics within that specific space. – timax Jan 23 '15 at 14:43
• @timax : $\:$ That "problem" with Euclid's axioms would not be a lack of completeness, it would be them entailing too much, i.e., it would be that they aren't sound. $\;\;\;\;$ – user57159 Jan 23 '15 at 14:58

I believe you are going about this the wrong way. The axiom you are trying to say is that some space exists to contain some points. However, by the mere existence of the points, we already have a space. The space is the collection of points and what the axioms allow. Nothing more, nothing less. If you agree that there are a bunch of points, then simply call the collection of points the set $\mathbb P$. $\mathbb P$ is our space, which is generally called the plane. The axioms define what you can do in that space.

The space is not where you put the points. The space is what is made from the points and the rest of the axioms.

You are correct in assuming that the axioms that you listed do leave a bit to be desired. Euclid's axioms are not exactly clear as to what certain things mean. A more usable form of the axioms of neutral geometry can be found here: http://www.math.washington.edu/~lee/Courses/444-5-2008/theorems-plane-geom.pdf

The existence and ruler Postulates may be what you want in a space.

The Existence Postulate: The collection of all points forms a nonempty set. There is more than one point in that set.

The Ruler Postulate: For every pair of points P and Q there exists a real number P Q, called the distance from P to Q. For each line $l$ there is a one-to-one correspondence from $l$ to $\mathbb R$ such that if P and Q are points on the line that correspond to the real numbers x and y, respectively, then PQ = |x − y|.

You've got to remember that this was done thousands of years ago. Euclid probably didn't realize the need for such an axiom. Euclid's Elements is one of the first attempts at making math logically rigorous. That took mathematicians hundreds of years to do and only really got going around the 1600s.

• Hilbert didn't define space or coordinates either, and that was a bit more recent. – Michael Biro Jan 23 '15 at 13:24

It's not possible to define space, but to define a model of space.

The axioms is for plane geometry, what can be drawn on a "paper". It could be extended by axioms how to construct three dimensional objects.

But still, space isn't a set of points.