I took a lot of Math courses, including a geometry grad course. So I’ve known all about Elliptic and Hyperbolic Geometry.
But recently I came across the Wikipedia article on Pasch’s axiom (1882). An “obvious” property of plane Geometry that Euclid even used, despite it not being stated as an axiom or proven from his axioms. (It can’t.)
In short: You run a straight line thru a triangle not hitting any vertex. If you cross one side you cross one other side. See the link for more info.
Pasch saw the problem and proved it’s independence of Euclidean Geometry.
I never heard about this. Later formalizations of Plane Geometry included it. It seems to be the same as something called the plane separation axiom.
Apparently “order” is the key, something which classic Geometry doesn’t pay much attention to.
What else did I miss that is of that flavor: so clearly obviously true but Euclid and others didn’t see it had to be stated?
Googling reveals surprisingly few pages on this. So clearly not as well known as the Big non-Euclidean Geometries.
I don’t know much about it, but from what I remember, Hilbert was really big on axiomatizing all the unstated assumptions that we tend to make in mathematics when working with object we’re familiar with. I recall in my first geometry class that some of the “axioms” we used were really quite vague, and all they seemed to say is that things work exactly as we expect them to given our real world experience with lines in the plane, and didn’t really provide a solid foundation. Whenever the “betweenness of points” axiom was used, it was mainly just “Look at the diagram - that’s how it has to be!” HIlbert tried (although I don’t know how successful he was) to provide a complete set of axioms which could logically derive all of Euclidean geometry without resorting to any handwaving. That plenty of other people are cited in the Wikipedia article working afterwards suggests HIlbert didn’t do it perfectly, and that there were a few handwavy things or unstated assumptions in some of the work he did.
As an example of why the concept of “betweenness” is important, see the proof that all triangles are isosceles. He looks at the case where R is between A and B and S is between A and C, and then he looks at the case where neither is between, but in the real world, for any triangle that isn’t isosceles, only one of them would be in between.
That seems like a weirdly specific axiom to me. Isn’t Hilbert’s I-2 axiom enough? For every two points there exists no more than one line that contains them both
If a line crosses one edge of a triangle, then it cannot exit on the same edge, else there would be more than one line that contains both points. And likewise, if it exits on another edge, it cannot reenter again on that edge. Can it then cross the third side? No, because then it would have to exit again, and on one of the two sides that it already crossed, again leading to a situation where you have two lines that contain the same two points.
There are degenerate cases of course, but I think in all of these you end up with the same lines anyway, and no clear notion of entering or exiting the triangle.
Perhaps another axiom is needed to ensure that the line cannot enter the triangle and never exit, perhaps by spiraling inward infinitely. “A line that enters a closed region must exit that region” or the like. But that seems more basic than Pasch’s axiom.
Why couldn’t the line continue on forever without hitting another side of the triangle?
The answer to this question (I think) is that the triangle’s sides divide the plane into two regions, the “outside” and “inside”. Proving that a closed curve did this was not easy. Jordan curve theorem - Wikipedia
It’s possible that the JCT actually has nothing to do with it, but I’ve always liked it as an example of something that’s intuitively obvious in plane geometry that is very hard to actually prove.
The Wiki page claims that proving the result for polygons is easy, and that the trouble is with “badly behaved” curves like fractals. Not sure if they’re just talking about Euclidean space, though. Still, that’s a pretty interesting theorem.
Which is one of the key points of Pasch’s Axiom being independent of Euclidean Geometry. I.e., you posit an axiom that a line can cross only one edge of a triangle and then show the resulting set of axioms is consistent. Weird, but consistent.
I clicked on the links to other axiomatizations like Hilbert’s and Tarksi* at the time I came across this. The usual Mathematical writing style stymies me. So it isn’t clear what the differences between these and classical Euclidean Geometry are.
E.g., “Hilbert’s axioms, unlike Tarski’s axioms, do not constitute a first-order theory because the axioms V.1–2 cannot be expressed in first-order logic.” And the significance of this is …?
Tarski has long been one of my favorite Mathematicians. Bonus points: his son-in-law is Andrzej Ehrenfeucht, the “games” guy.
Consider the triangle where one side is the segment (x,0) for -1 <= x <= 1, the second side is the segment (1,y) for y >= 0, and the third side is the segment (-1,y) for y >= 0. In this triangle, the line x = 0 intersects one and only one side of the triangle.
Wait, what’s that you say? The triangle I described isn’t actually a triangle? Prove it.
Hilbert’s formulation of the axiom doesn’t mention triangles, and doesn’t apply to your example since it fails the setup:
I don’t see any point C, or segments AC or BC.
I have one that works, though: consider just a cylindrical geometry. Put three points nearly (but not quite) along a great circle. That’s a “triangle”. A fourth line that spirals up like a barber pole will hit one edge but no other.
Still, it seems that there are better ways to fix this problem… I dunno.
Well, I think the burden of proof is upon you to prove that it is a triangle. This is geometry after all, I’m not taking your word for it.
And
I only see two vertices in the figure you’ve described; additionally, wikipedia defines an ‘edge’ as (inter alia) a line segment, of which there is only one in your figure. So, yeah, I am unconvinced this is a triangle.
Though I admit I’ve only thought about Pasch’s axiom enough to only dimly, if at all, understand the need for it. Can anyone describe the geometry where Pasch’s axiom is false? (Would a torus be an example? A triangle goes roughly along the top of the doughnut, with a line forming a loop through the middle: the line doesn’t end anywhere, and goes through the triangle only once)
That works, as does the cylindrical geometry I mentioned earlier.
Also: any Euclidean geometry with a cutout. Suppose I take a normal Euclidean plane and just chop out a hole at x^2+y^2<=1. Position a triangle so that one of the edges crosses the hole. Now I can place a line that crosses an edge to get into the triangle but slips out via the hole.
That actually fails the parallel postulate, though. I think all of the counterexamples so far fail some other axiom, so I still don’t have a case where Pasch is actually needed.