The Half-life of Facts.

Non-euclidian geometry is simply about other types of spaces, developed using different sets of axioms.It's not in disagreement with Euclidian geometry, but a generalisation of it.

The fifth, or parallel postulate, is the major difference between generalized geometries.

The fifth postulate was assumed to be axiomatic for almost a millennia prior to the generalized development of the non-Euclidian forms in the early 19th Century.

Interestingly a similar question about the Algebra was also occurring at the same time when the axiom of multiplicative commutation was also questioned. A question that lead (in part) to the development of Modern (abstract) Algebra, number fields and groups.

Rap

Eh .... no. If you look at the Wikipedia page about noneuclidean geometry, you will find that it emerged from a disagreement within mathematics. When non-Euclidean geometries were first proposed, Conservative mathematicians (for lack of a better adjective) rejected them as invalid. It may not appear this way to us today, but at the time there was a genuine disagreement among qualified mathematicians about mathematics. That is enough to support my point that you can disagree about mathematics.

I read something once about the definition of a straight line.

The shortest distance between two points is a straight line, according to Euclidean geometry.
But in space, it is possible for the shortest distance between two points to be a curved line, because space curves.
Then we have the paradox of a curved, straight line. This leads to other things such as the possibility of triangles where the total sum of the angles can be more or less than 180 degrees.

But then again, if you draw a straight line on a piece of paper, then make the paper curved, is the line still straight?

I agree that it is absolutely possible to disagree about mathematics.

Think of a geometry on the surface of a sphere (the simplest two dimensional model of an elliptic geometry). A straight line in this geometry then is the arc cut by a plane through the center of the sphere (a great circle arc). This straight line is the shortest distance between two points on the surface of the sphere.

Moreover, a triangle created by three straight lines in this geometry has a total angle sum of more that 180 degrees (pi radians).

BTW there are no parallel lines in elliptic geometry (fifth postulate).

This simple model is also the basis to celestial navigation and spherical trigonometry.

Rap

Yes, I thought of drawing triangles on the surface of the earth, for instance.

On that surface, it seems to me that what constitutes a straight line depends on how we define it.
If you projected a beam of light from somewhere on the equator, along the surface of the planet, intending to hit a point on the north pole, you would be disappointed. The beam moves in a straight line.
However, if you were to travel the distance in an airplane, at a fixed altitude, and you aim the plane in the same direction you aimed the beam of light, sooner or later you would reach the north pole. The plane moves in a straight line.

And yet, the beam of light and the plane, both moving in straight lines and starting at the same place, end up in different places.

Pick any two points on the surface of a sphere--now draw a path between those two points. How many different paths can you draw?

I'll answer this --you can draw an infinite number of paths--so how can you determine which path is the situational definition of a line--it is the shortest one and that path is unique. It is also a great circle arc on the surface of that sphere.

Straightness is not a elliptic (or hyperbolic) definition for a line in this geometry--the shortest path is.

In planar (Euclidian) geometry that shortest distance path is a straight line. In non-Euclidian geometry that shortest path is not necessarily straight from a planar perspective.

BTW I recall seeing this Analysis proof in a long ago class for geometries in general.

Rap

The apparent contradiction is easily resolved: the two straight lines do not belong to the same space. The plane belongs to and goes along the curved space of the atmosphere, while the laser is part of a much wider space, the universe. No wonder they don't arrive in the same place.

There's no disagreement or dispute between mathematicians studying non-euclidian spaces and others studying euclidian spaces. There might have been at start but not anymore. These are not two 'camps' like Marxists and anti-Marxists. The same person can easily work on different kinds of space. They apply to different situations too.

So of course two students or professors of mathematics can temporarily argue on a proof, but there is an objective solution to their dispute, an objective solution which is in grasp, either already found and published, or if not, it will present itself to someone at some point, be reviewed and published, after which the proof will become universally accepted.

There is a strong universal dimension to mathematics, which is good news. It's like a proof of the universality of reason.

This is interesting.
But it is a matter of perspective, as I see it.
I asked earlier; if you draw a line on a plane that is the shortest distance between two points, that's a straight line.
If you then make the plane curved, would you say that the line is still straight? It still follows the shortest distance between the two points.

I am thinking of light traveling past a strong gravitational force that curves space around itself. The idea, if I understand it correctly, is that gravity warps space itself, so that the shortest distance between two points, while still being a straight line because it is space that is curved, will appear curved when viewed from outside of that gravitational influence.

Yes, precisely. So it depends on your subject matter which geometry is relevant.

Yes you can, occasionally, and only until a workable proof or synthesis is found. One may argue that other sciences too are universal, at least they aim to be, but none is as universal as mathematics.

Isn't that true of anything? You can only argue until agreement is reached.

Except it's much easier to disagree about the existence of God, history or global warming than about algebra.

Is that a fact?

Joefromcanada: what a silly question! Shame on you.

Sit in the corner and be quiet, Millerqueenoftheghetto, adults are talking.

It is a "fact " in the constructivist sense of the word.

Perhaps I shouldn't be so dismissive without offering an alternative version of constructivism.

Decades ago, I heard Bavarian Public Radio interview some Austrian psychologist. He accepted the premise of constructivism that reality is an artefact constructed in part by the individual, in part by society. But he also added a qualification. My leaky memory remembers him saying something like this:

"When a lumberjack, a poacher, and a pair of lovers walk into a forest, they'll come back out having constructed very different realities about it. But they all come back out. They could, instead, have constructed realities in which bears are safe to pet, rotten tree branches are safe to climb onto, and so forth. As a result, they could have ended up being carried out of the forest. Then the realities they'd constructed would have died with them, revealed as what naive realists call 'wrong'."

I have no problem with this variant of constructivism.

I edited my last post, thinking I was quoting myself. I'm sorry about any confusion I caused.


What I originally had written, and thought I was quoting, is this:


In the constructivist sense of the word, a "fact" is what people agree on. But people don't agree whether "[t]he 'thingness' of the blob is a function of its interaction with the 'thinger', not an independent existential state." Therefore, your statement wasn't a fact even by the low standards of (your version of) constructivism.

That doesn't answer my question. I didn't ask whether your statement was a "fact," but whether it was a fact. Or are you saying that there are no such things as facts, only "facts?"