Yeah, pretty much. I thought the definition of circumscribing a triangle would be finding a circle which goes through the three vertices of the trangle, and which contains the triangle. I did click through to circumscribed on the site, but that entry is waiting to be written.
Sounds like a perfectly good definition to me. It’s just that in the hyperbolic plane it’s possible to find three points which aren’t collinear but which also don’t lie on any common circle. Generally, three points which are “nearly collinear” will not have a circumcircle.
What is true both in the Euclidean and hyperbolic planes is that any three distinct points lie on a unique curve of constant curvature. In the Euclidean plane, there are only two types of curves of constant curvature, lines and circles, and since the vertices of a triangle aren’t collinear they must lie on a common circle instead. But in the hyperbolic plane there are four types of curves of constant curvature: lines, circles, horocycles, and the type of curve I described in case (c) which doesn’t have a name that I’m aware of. The vertices of a triangle aren’t collinear in the hyperbolic plane any more than in the Euclidean plane, but that still leaves three other cases.
I think, given the poster’s admission of non-math-ness, that this question deserved to be addressed sans jargon. Forgive me for retreading some previous posts.
Euclid was attempting to unify and extend what was known about geometry in his day. The postulates reflect what was known at the time, and are described as a foundation from which to build the rest of geometry.
Postulate 1 simply defines what a line is, something that connects two points
Postulate 2 basically declares that a line need not end at the two points required to define it.
Postulate 3 defines a circle.
Postulate 4 simply reinforces that equal things are equal (although I’d be interested in a geometry that denies the FOURTH postulate somehow).
From these four, Euclid would hope to be able to derive everything that could be known from geometry. However, using these four, he could not derive the fifth. However, the idea of there being a single parallel line to a given line through a given point comes in quite handy for proving other things, so Euclid basically gave up and put in the ungainly Fifth Postulate.
Later, it was discovered that the Fifth postulate need not be true. For example, the Earth is not flat, but curved into a sphere. If you pick two points on the surface of the earth, and draw an “infinite” line through them (postulates 1 and 2), you get a Great Circle about the Earth, which becomes your de facto straight line in this context. If you then pick a point that is not in this line, you find that you can not draw any straight line through the point that does not intersect the first one. But everything else you can come up with on the surface of a sphere without requring postulate five to be true is consistent. There are also geometries that are consistent on surfaces curved in such a way that you can draw more than one non-intersecting lines through the third point
On a flat surface, you have one parallel line through the non-linear point. On non-flat surfaces, you have zero parallel lines or multiple parallel lines. Between the two you have a complete understanding of geometry based on the other four postulates. This has proven fruitful in a number of fields. These days, the fifth postulate is effectively not a postulate except you are deliberately restricting your discussion of geometry to flat surfaces.
HOWEVER: If Mr. Matta’s proof is found to be correct, then it means that the Parallel postulate is true in all contexts, not just flat surfaces, and all of the geometries of curved surfaces, many of which have led to important real-world results in a number of areas, are based on a falsehood. Many of the mathematical models by which we believe we understand the Universe at this point will have been shown to be wrong. Math and Physics will be shaken to their core, and we will be back a square one.
He’s probably wrong, however.
One nitpick: “points” on the surface of a sphere are pairs of antipodal points. There are an infinite number of great circles through such a pair, so they don’t uniquely determine a “line”.
Even if you want to define a “point” on a spherical surface as a pair of antipodal points (in the conventional flat-surface sense), in the same way that we define “straight line” as a great circle about the sphere, then two “points” will still have a unique “straight line” that connects them (i.e., there will be one great circle that passes through all four points, given that they are two pairs of antipodal points), preserving consistency with flat Euclidean geometry.
Not quite. “Equal things are equal” (or, more explicitly, two things each equal to a third thing are themselves equal) is one of the Common Notions of Euclid. Strictly speaking, these are five more postulates, but Euclid considered them to be of such a “duh” nature that he didn’t distinguish them by calling them such. And some of them turn out to not be quite as obvious as he thought: For instance, another Common Notion was that the whole is equal to the sum of its parts, which is difficult to reconcile with the Banach-Tarski Theorem (currently being discussed in another thread).
What the fourth postulate actually says is that all right angles are equal, and it is certainly possible to construct geometries where this is not the case. For instance, on the surface of a cone, a right angle with vertex at the vertex of the cone will be smaller than a right angle anywhere else on the cone. In fact, I’m pretty sure that it’s not true in General Relativity, either, since it’s the only assumption of homogeneity in Euclid’s postulates, and space in GR is not in general homogeneous.
scotandrsn, I think that ultrafilter’s point was that antipodal points must be considered a single point, else one could try to define the unique line between two antipodal points. But there isn’t a unique line there, because that pair of points is really only a single point.
That’s the crucial point. I was visualizing triangles close to equilateral. Thanks.
Well, one problem with working without jargon is that a lot of the really deep stuff is completely lost. For one thing, it’s difficult to think of the purely formal nature of Euclidean-style geometry without jargon.
Hilbert famously stated that the proofs of geometry should hold true if everywhere (including in the definitions) the words “point”, “line”, and “plane” are replaced by “table”, “chair”, and “beer-mug”. More into the real world (well, the math world), throw out Euclid’s definitions and just take the postulates as things which need to be satisfied. Now, many such models (collections of things and relations between them satisfying the postulates) don’t consist of what we intuitively think of as “points”, “lines”, and “planes”, but all that we ask is that the postulates are satisfied and the rest follows from logic.
To put the catastrophe that would result if this guy were right into these terms, given a collection of things satisfying the first four postulates, he says we can prove that the fifth one is satisfied as well. The problem is that we can use this collection to build another collection satisfying the first four (possibly redefining some of our terms and relations) which violates the fifth. But his proof shows that the fifth must be satisfied! Therefore the first four contain a contradiction, and there are no collections which satisfy the first four postulates.
Now here’s the problem: if we assume that another branch of math called set theory has a model (which everyone does), we can build a model of Euclidean geometry (a collection of things and relations satisfying the five postulates). But we just showed that no such things exist, which means that there can’t be any models of set theory. In that case, pretty much all mathematics (and mathematical descriptions of, well, anything) crumbles.
Why do you consider a proof which shows that mathematics is inconsistent to be a bad thing? Down with consistency! Viva la revolution! Smash the logical oppressors! Let “2 + 2 = 5” be our motto! Freedom! Freedom! Freedom!
Only for very large values of 2. 
May I ask what you do for a living, so I may ridicule it and hope that your career is rendered obsolete?
Forget that, actually. Note what a vast portion of Maryland’s solvency is dependant upon mathematics (if only through the NSA: the single largest employer in the state) and consider what your life would be like if that went under.
Maybe I’m not reading this post the way you meant it, but calm down. I’m sure he didn’t mean that he hopes the principles of math are proven inconsistent. He was merely making a light-hearted comment. Nothing to get worked up over.
snark in, snark out
Works with garbage too.
Could we go back to Peano’s axioms? (I like to be prepared. 10 independant proofs! Sure, one or two might be wrong, but what are the odds they’ll all be wrong? ;))
Peano’s axioms follow from the axioms of set theory.
Wendell is a mathematician. Does the concept of a joke elude you?
But that wouldn’t mean Peano’s axioms were wrong. Sort of:
Set Theory -> Euclidean Geometry means NOT(Euclidean Geometry) -> NOT(Set Theory)
but Set Theory -> Peano’s Axioms doesn’t mean NOT(Set Theory) -> NOT(Peaono’s Axioms)
Can you obtain either Set Theory or Euclidean Geometry from Peano’s Axioms?
Note the later comment in re snarkiness.
The problem is that there’s not a lot you can do with arithmetic without throwing in a logical background in which you could construct models of ZF.