# Why is “points exist” not an axiom in geometry?

I am not sure why "points exist" is not an axiom in geometry, given that the other axioms are likewise primitive and seemingly as obvious.

• Why not "lines exist" then points would be derived objects? Not an answer, just a thought. – Paul Mar 1 at 9:05
• If the question is about Euclid's Elements specifically, there's alot missing in those axioms. For example, in the very first proof, Euclid assumes that two circles draw at a certain distance from one another must intersect. – Jack M Mar 1 at 9:50
• See David Hilbert, The Foundations of Geometry (1899), page 2 : "Let us consider three distinct systems of things. The things composing the first system, we will call points and designate them by the letters $A, B, C$ ; those of the second, we will call straight lines and..." – Mauro ALLEGRANZA Mar 1 at 12:09
• Defining too much detail can quickly devolve into a problem wonderful put by Lewis Carrol. en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles platonicrealms.com/encyclopedia/Carrolls-Paradox – Duck of Death Mar 1 at 22:02
• I want to say something more about what @JackM said. Euclid's Elements is not only too imprecise to be considered a proper axiomatization of geometry, it is also woefully incomplete and Euclid made numerous unfixable errors, contrary to popular accounts. It is not his fault, since his attempt was a very commendable one (hardly anyone can claim to have even tried before him). But it would be foolish for any student today to use his Elements as a place to learn rigorous Euclidean geometry. – user21820 Mar 2 at 9:46

In the presentation I have most convenient to me (Hilbert's axioms), the axioms for plane geometry start with a trio of "axioms of incidence". One of those axioms is "There exist three non-collinear points". That axiom certainly includes the existence of points.

• +1. For a specific reference, see Wikipedia's "Hilbert's Axioms" entry. In that listing, the third Incidence axiom reads "There exist at least two points on a line. There exist at least three points that do not lie on the same line." Further, the eighth Incidence axiom reads "There exist at least four points not lying in a plane." – Blue Mar 1 at 8:55
• The version I'm working from (Hartshorne's textbook) organizes it a little differently. The other two incidence axioms in that version are "For any two points $A,B$, there exists a unique line $l$ containing them", and "Every line contains at least two points". (The axioms from #4 on in the Wikipedia version are axioms for three-dimensional geometry) – jmerry Mar 1 at 9:06
• I still would find it interesting to know why it's left out in so many places then. – user10869858 Mar 1 at 9:12
• @jmerry: Probably every text organizes things a little differently. :) I just thought OP would benefit from access to some version of a "complete" reference, in case there are additional questions about what a comprehensive axiomatic system may-or-may-not cover. – Blue Mar 1 at 9:12
• @user10869858: What do you mean by "so many places"? Note that Euclid's Elements itself is notoriously lacking in covering all the necessary logical bases; on the other hand, Hilbert's axioms may overwhelm a new learner. It's unsurprising (though perhaps a little unfortunate) that introductory treatments (say, high school textbooks) might make certain compromises, confident that advanced courses and/or readily-available resources (like Math.SE! ;) will fill in any gaps. – Blue Mar 1 at 9:24

Getting a bit meta here, but the point of axioms is not just to play pedant. They meaningfully differentiate between scenarios that might be considered. If the geometry of a system that didn't have points was a thing, then maybe there would be a field with an axiom of this nature.

Side note: often the mathematical fields of interest to the outside world are those that mimic something in real life. There, it's obvious that "the triangle equality holds". However in maths it is not clear that would always be the case. Indeed there are interesting cases in which it doesn't hold, and these need to be differentiated.

EDIT below, some extra thoughts:

Often the 'existence' axioms are not needed for the simple grammatical reasons. It's often the case that arguments go (or can be rephrased) something along the lines of: the following restrictions exist/all things considered must obey these rules/there are no things that don't obey these rules. If this is a true statement without the caveat of non-emptiness, then it is in the general case as the empty case is automatically true. This is often the easier to work, and doesn't need 'points exist' as an axiom.

• You can't just say an axiom is unnecessary because the case where it doesn't hold is boring. – user2357112 Mar 1 at 17:54
• Strongly disagree with your first sentence. Axioms have various purposes (and the term "axiom" is used with different meanings), but one of their most important purposes is to "play pedant" and provide a thorough logical foundation for a subject. This is certainly the case for the traditional use of axioms in geometry. – Eric Wofsey Mar 1 at 19:44
• @user2357112 I think there is a miss confusion here. I am not saying that at all, I am saying you can't omit them if the the case it doesn't hold is interesting. (en.wikipedia.org/wiki/Contraposition) – ANone Mar 4 at 9:32
• I am wondering if you meant "nothing for which it doesn't hold is interesting" to be included in "[w]hat's not to say". – hkBst Mar 5 at 10:48
• @hkBst I think I stand by what I said, but as you are not the first to comment, I will change the wording. – ANone Mar 5 at 11:03

One point that is missing from the other answers is that of vacuous truth. If formulated properly, we do not need points for the theorems in geometry to hold. If there are no right angled triangles then the Pythagorean theorem trivially holds for all of them, if there are no circles, Thales' theorem hold for all of them and so on. There might be a few theorems that cannot be restated in the form "For all objects x of class Y we have that ...", but even those could be reformulated as "If there exists a point, then...".

So, say you take a model-theoretic point of view of geometry and look at it just as a "set of theorems that hold for any model which satisfies those axioms". Then adding "points exists" as an axiom will only exclude the empty set as a model, which was kind of a boring model anyway. There might be a reason to do so for convenience and readability, in the same way as for example most authors exclude $$\{0\}$$ from the fields, but the differences in the resulting theory will be rather minor.