Thread: Mathematical Numbers are Theoretical; Not Proven

That is not the definition. What you just said applies to skew lines.

See above.

You said the rules are disproven and outdated. Did you change your mind? Are the opposite angles of a parallelogram congruent or not?

If mathematics is not "real", why are you arguing for the truth of hyperbolic geometry?

Don't make things up. When did I ever say anything along the lines of the rules of Euclidian geometry being outdated or 'disproven'?? It was discovered a century ago that Euclidian geometry doesn't apply to the universe, but Euclidian geometry is still completely valid in itself, and all the proofs are flawless. The only issue is that the axioms - specifically the parallel axiom - aren't true in the universe. So all of the proofs built upon those axioms also don't apply to the universe. They're completely valid within Euclidian geometry though, and have many applications elsewhere, such as when working with complex numbers.

I'm not arguing for the truth of hyperbolic geometry either; again, hyperbolic geometry is only true within the axioms of hyperbolic geometry. The geometry of the universe isn't actually hyperbolic either, it's more complicated than that.

So one angle of a square is 89 and a whole bunch of decimal places while its opposite angle is 90 and a whole bunch of zeros and then another digit after the decimal?

I don't see why not (although they needn't sum to 180 in case that's what you were getting at). It depends on the local amount of curvature. If you're near a very large dense mass such as a black hole the angles could be vastly different.

All four angles of a square? Then what would make it a square?

A square is a polygon with four sides of equal length and four equal angles. Here is what a hyperbolic square looks like:
http://upload.wikimedia.org/wikipedi...olic_plane.png

Originally Posted by Xei

See above.

See specifically what above?

Originally Posted by Xei

Don't make things up. When did I ever say anything along the lines of the rules of Euclidian geometry being outdated or 'disproven'??

Originally Posted by Xei

That hypothesis is outdated by about a century now.

You said that in response to my question about whether opposite angles of a parallelogram are congruent. It is in post #61.

Originally Posted by Xei

It was discovered a century ago that Euclidian geometry doesn't apply to the universe, but Euclidian geometry is still completely valid in itself, and all the proofs are flawless.

Earlier, you said they are nothing more than good approximations.

Originally Posted by Xei

The only issue is that the axioms - specifically the parallel axiom - aren't true in the universe. So all of the proofs built upon those axioms also don't apply to the universe. They're completely valid within Euclidian geometry though, and have many applications elsewhere, such as when working with complex numbers.

The second dimension is in the universe.

Originally Posted by Xei

I'm not arguing for the truth of hyperbolic geometry either; again, hyperbolic geometry is only true within the axioms of hyperbolic geometry. The geometry of the universe isn't actually hyperbolic either, it's more complicated than that.

How is it true within itself if it is just something humans made up? Does it have any more reality than truth within The Wizard of Oz? What is the difference between math and science fiction, in terms of truth? The Death Star is real within Star Wars.

Originally Posted by Xei

I don't see why not (although they needn't sum to 180 in case that's what you were getting at). It depends on the local amount of curvature. If you're near a very large dense mass such as a black hole the angles could be vastly different.

Then it would not be a square.

Originally Posted by Xei

A square is a polygon with four sides of equal length and four equal angles. Here is what a hyperbolic square looks like:
http://upload.wikimedia.org/wikipedi...olic_plane.png

That makes no sense. It is not even a polygon, much less a square. A polygon is completely enclosed, unlike that figure, and the sides of a polygon are segments, which are straight.

See specifically what above?

My post to Licity who made the same (fair) point.

You said that in response to my question about whether opposite angles of a parallelogram are congruent. It is in post #61.

No, I said, "Space isn't Euclidian. That hypothesis is outdated by about a century now."

Which is exactly what I've been saying all along. Do you still not understand the distinction?

Euclidian geometry is only true within the axioms of Euclidian geometry; the same condition applies to all branches of mathematics. In reality, however, space is not Euclidian; it is warped by mass.

However that does not make Euclidian geometry redundant, for two important reasons:

- Euclidian geometry is an incredibly accurate approximation on Earth so engineers etc. don't even have to worry about it.
- The axioms of Euclidian geometry can be applied perfectly to other areas of mathematics, such as vectors, the Argand plane, etc.; other areas of maths which can then either be applied to the real world as models (with varying degrees of accuracy), or studied for their own sake (pure mathematics).

Earlier, you said they are nothing more than good approximations.

Nope I'm still repeating exactly the same things for the tenth time. Read more carefully I guess.

The second dimension is in the universe.

I can't glean any sense from this at all... if you take any 2D plane through the universe, it will be non-Euclidian, because it's a cross section of a 4D non-Euclidian space.

How is it true within itself if it is just something humans made up? Does it have any more reality than truth within The Wizard of Oz? What is the difference between math and science fiction, in terms of truth? The Death Star is real within Star Wars.

Well exactly. Humans 'made it up'. Maths is not a physical/objective entity. The axioms and resultant models are often extremely good approximations to the real world though so they do have practical uses.

That makes no sense. It is not even a polygon, much less a square. A polygon is completely enclosed, unlike that figure, and the sides of a polygon are segments, which are straight.

It's a representation of hyperbolic space, not Euclidian. Those lines are straight within hyperbolic axioms.

The little bits in the corners aren't part of the square.

I don't feel like hunting down another one of your quotes, but you did say that Euclidian geometry is taught and required everywhere because it provides for great "approximations". You also denied and denied that the opposite angles of a parallelogram are congruent and said the idea that they are is outdated. That was the point of your response I quoted, which was in response to my point that the opposite angles of a parallelogram are congruent. I don't think you were talking to yourself when you said it.

The second dimension is part of the multidimensional system of our universe, but that does not mean it starts acting like some greater dimension. The opposite angles of a parallelogram are congruent, no matter what you say happens in greater dimensional aspects.

If these systems of math are fiction ideas that are only true within themselves, why do they provide for such astoundingly good "approximations" concerning reality?

Good "approximations" on small scales. The curvature of space by gravity becomes increasingly irrelevant as the scale reduces.

If I draw a line on a sphere, then draw a line at 90° to it, and another one 90° to that line, are the first and last lines parallel? Do they intersect? The answers are not the same as in Euclidean geometry (on a flat plane). This is because a sphere is curved. But if you take a small enough piece out of a large sphere, it looks completely flat as far as a human eye is able to see, and Euclidean axioms hold as far as we can see by experiment. Now generalise that to 3D (or more dimensions).

Originally Posted by Universal Mind

I don't feel like hunting down another one of your quotes, but you did say that Euclidian geometry is taught and required everywhere because it provides for great "approximations". You also denied and denied that the opposite angles of a parallelogram are congruent and said the idea that they are is outdated. That was the point of your response I quoted, which was in response to my point that the opposite angles of a parallelogram are congruent. I don't think you were talking to yourself when you said it.

The second dimension is part of the multidimensional system of our universe, but that does not mean it starts acting like some greater dimension. The opposite angles of a parallelogram are congruent, no matter what you say happens in greater dimensional aspects.

If these systems of math are fiction ideas that are only true within themselves, why do they provide for such astoundingly good "approximations" concerning reality?

In reality, the opposite angles of a parallelogram are not equal, because reality is not Euclidian. I have never denied that they are equal within Euclidian geometry - indeed I asked you to prove it - the point is that the physical universe is not actually Euclidian.

There aren't really any objective 2 dimensions 'out there', but if you were to take a 2D cross section of the universe, it would also not be Euclidian. Standard hyperbolic geometry is 2D. It is really just a human assumption that Euclidian geometry and not some other geometry should apply well to larger scales than the ones we percieve; a completely understandable one, but ultimately a flawed one. It is just like how it was assumed for a few hundred years that Newtonian mechanics was objectively true; turns out that at small velocities it is only an approximation, and at very large velocities it is pretty much useless.

Euclidian geometry is an extremely good approximation on Earth because the axioms are almost perfectly true. It's only the parallel axiom which is incorrect, and even then, only by a tiny amount. The gravitational field of the Earth is really extremely weak, and nowhere near strong enough to cause geometry to diverge from the Euclidian model by a significant amount. Redfishbluefish explained it quite nicely with the paper analogy.

So (both of you), if one angle of a parallelogram is 70 degrees, what is its opposite angle? I am not asking for what it almost is. I am asking you what it is. You might need to answer that with a range, based on what I know about your view.

In which geometry?

In Euclidian, it is 70 degrees.

In hyperbolic, I am not 100% sure about this, but I think it could be any n ∈ ℝ+, depending on the amount of local curvature.

In reality, as in, if you actually made a parallelogram out of bits of wood or something, or if you were just talking about any abstract lines in space (you could define them by the path taken by a photon), I think you get the same result as above. It depends entirely on the amount of local spacetime curvature. It would only be 70 if you were in some volume of space not affected by gravitation, and as far as we know there is no such location.